J. Math. Soc. Japan Vol. 53, No. 4, 001 Gap modules fo diect poduct goups Dedicated to Pofesso Masayoshi Kamata on his 60th bithday By Toshio Sumi (Received Nov. 4, 1999) (Revised Jun. 7, 000) Abstact. Let G be a nite goup. A gap G-module V is a nite dimensional eal G-epesentation space satisfying the following two conditions: (1) The following stong gap condition holds: dim V P > dim V H fo all P < Ha G such that P is of pime powe ode, which is a su½cient condition to de ne a G- sugey obstuction goup and a G-sugey obstuction. () V has only one H- xed point 0 fo all lage subgoups H, namely H A L G. A nite goup G not of pime powe ode is called a gap goup if thee exists a gap G- module. We discuss the question when the diect poduct K L is a gap goup fo two nite goups K and L. Accoding to [5], if K and K C ae gap goups, so is K L. In this pape, we pove that if K is a gap goup, so is K C. Using [5], this allows us to show that if a nite goup G has a quotient goup which is a gap goup, then G itself is a gap goup. Also, we pove the convese: if K is not a gap goup, then K D n is not a gap goup. To show this we de ne a condition, called NGC, which is equivalent to the non-existence of gap modules. 1. Intoduction. Let G be a nite goup and p a pime. In this pape we assume that the tivial goup is also called a p-goup. We denote by P p G a set of p-subgoups of G, de ne the Dess subgoup G fpg as the smallest nomal subgoup of G whose index is a powe of p, possibly 1, and let denote by L p G the family of subgoups L of G which contains G fpg. Set P G ˆ6 p P p G and L G ˆ6 p L p G : Let V be a G-module V. We say that V is L G -fee, ifv Gf pg ˆ 0 holds fo any pime p. Set D G as a set of pais P; H of subgoups of G such that P < Ha G and P A P G. We denote by D G be a set of all elements P; H of D G with P B L G. Clealy note that this set equals to D G if P G V L G ˆ q holds. We de ne a function d V : D G!Z by d V P; H ˆdim V P dim V H : 000 Mathematics Subject Classi cation. 57S17, 0C15. Key Wods and Phases. gap goup, gap module, eal epesentation, diect poduct.
976 T. Sumi We say that V is positive (esp. nonnegative, esp. zeo) at P; H, ifd V P; H is positive (esp. nonnegative, esp. zeo). Fo a nite goup G not of pime powe ode, a eal G-module V is called an almost gap G-module, if V is an L G -fee eal G-module such that d V P; H > 0 fo all P; H A D G. If L G V P G ˆq holds, an almost gap G-module is called a gap G-module. We can stably apply the equivaiant sugey theoy to gap G-modules. We say that G is a/an (almost) gap goup if thee is a/an (almost) gap G-module. A nite goup G is called an Olive goup if thee does not exist a nomal seies P / H / G such that P and G=H ae of pime powe ode and H=P is cyclic. A nite goup G has a smooth action on a disk without xed points if and only if G is an Olive goup, and G has a smooth action on a sphee with exactly one xed point if and only if G is an Olive goup (cf. Olive [7] and Laitinen-Moimoto [3]). It is an impotant task to decide whethe a given goup G is a gap goup. In fact, if a nite Olive goup G is a gap goup, then one can apply equivaiant sugey to convet an appopiate smooth action of G on a disk D into a smooth action of G on a sphee S with S G ˆ M ˆ D G, whee dim M > 0 (cf. Moimoto [4, Coollay 0.3]). Laitinen and Moimoto [3] de ned the G-module V G ˆ R GŠ R 0 R G=G fpg Š R ; p which is useful to constuct a gap G-module, and poved that a nite goup G has a smooth action on a sphee with any numbe of xed points if and only if G is an Olive goup. This G-module also plays an impotant ole in this pape. The pupose of this pape is to study the question when a diect poduct goup is a gap goup. The main theoem of this pape concens a diect poduct K D n, whee D n is the dihedal goup of ode n fo nb1 D ˆ C and D 4 ˆ C C. Theoem 1.1. Let n be a positive intege and let K be a nite goup. Then K is a gap goup if and only if G ˆ K D n is a gap goup. This pape is a continuation of ou joint wok with M. Moimoto and M. Yanagihaa [5]. The key idea of the poof can be found in [6]. In [5, Theoem 3.5], we have shown that if P K V L K ˆq and K C is a gap goup, so is K F fo any nite goup F. In Lemma 5.1, we show that if K is a gap goup, so is K C, which is the case whee n ˆ 1 in the main theoem. Using Lemma 5.1 and [5, Theoem 3.5], we obtain the following theoem. Theoem 1.. If a nite goup G has a quotient goup which is a gap goup, then G itself is a gap goup.
Gap modules fo diect poduct goups 977 Recall that G is an Olive goup if it has a quotient goup which is an Olive goup. The oganization of the pape is as follows. In Section, we estimate d V P; H fo a K L -module V by chaactes of ieducible K- and L- modules. In Section 3, we nd a gap G-module fo a cetain diect poduct goup of symmetic goups. The goups S 4 and S 5 ae not gap goups but S 4 S 5 is a gap goup. In Section 4, we intoduce a condition NGC and show that G holds NGC if and only if G is not a gap goup. We de ne a dimension matix and give the condition equivalent to one being a gap goup by using a dimension submatix. In Section 5, by using the esults in Section 4, we show that K C is an almost gap goup if so is K. In Section 6, we show that thee ae many nite goups G such that P G V L G ˆq holds but G ae not gap goups. As an application we completely decide when a diect poduct goup of symmetic goups is a gap goup. Since a gap goup which is a diect poduct of symmetic goups is an Olive goup, it can act smoothly on a standad sphee with one xed point.. Diect poduct goups. Let G ˆ K L be a nite goup. We denote by w V the chaacte fo a G- module V. Let P and H be subgoups of G such that H : PŠ ˆ. Then :1 d VnW P; H ˆ 1 w jpj V p 1 x w W p x w jhj V p 1 y w W p y x A P ˆ 1 jpj w V p 1 h w W p h ; h A HnP y A H whee V (esp. W )isak- (esp. L-) module and p 1 : G! K and p : G! L ae the canonical pojections. We set and D G ˆf P; H A D G j H : PŠ ˆ HG fg : PG fg Šˆand PG fqg ˆ G fo all odd pimes qg: D G ˆD G V D G : Then d V G is positive on D G nd G. We have shown a estiction fomula that eads as follows:
978 T. Sumi Poposition. (cf. [5, Poposition 3.1]). Let K be a subgoup of an almost gap goup G such that G fg < Ka G. Then K is an almost gap goup. Futhemoe, if the ode of G fg is not a powe of a pime, then G fg is an almost gap goup. Let RO G L G be an additive subgoup of RO G geneated by L G -fee ieducible eal G-modules. Thee is a goup epimophism j : RO G! RO G L G which is a left invese of the inclusion RO G L G,! RO G. Fo a G-module V, we set V L G ˆ j V. Then V L G is an L G -fee G-module and :3 V L G ˆ V V G 0 V V G Gf pg : pjjgj holds. In paticula, V G ˆR GŠ L G holds. Hee the minus sign is intepeted as follows. Fo some intege l > 0, we egad V as a G-submodule of lr GŠ with some G-invaiant inne poduct. Fo a G-submodule W of V, we denote by V W the G-module which is othogonal complement of W in V. Fo distinct pimes p and q, V Gf pg V V Gfqg ˆ V G holds, since G fpg G fqg ˆ G. Then the diect sum of V V G Gf pg is a G-submodule of V V G. The following is a estiction fomula fo an odd pime p. Poposition.4. Let K be a subgoup of G such that G fpg < Ka G fo a pime p. Suppose thee is a nomal p-subgoup L of G such that LK ˆ G. If G is a gap goup then so is K. Poof. Let W be a gap G-module. Since W K ˆ 0, we set V ˆ Res G K W L L K. We show that V is a gap K-module. It su½ces to show that V is positive on D K. Let P; H A D K. Then P is a p-goup and thus LP is also a p-goup. Theefoe it follows that LP; HP A D G and d V P; H ˆ d W LP; HP P q d W LPK fqg ; HPK fqg ˆd W LP; HP d W LPK fg ; HPK fg. We claim that LK fg ˆ G fg and thus d V P; H ˆd W LP; LH > 0. Fo g ˆ lk A G ˆ LK, we obtain g 1 LK fg g ˆ k 1 Lk k 1 K fg k ˆLK fg. Hence LK fg is a nomal subgoup of G. Clealy LK fg a G fg. / K ƒƒƒ? y / G ˆ LK ƒƒƒ K V G fg / ƒƒƒ? y G fg ˆ L K V G fg / ƒƒƒ K fg? y LK fg Since L V K V G fg ˆ LVG fg V K ˆ L V K ˆ L V K fg, it follows that G fg : K V G fg Šˆ LK fg : K fg Š. Theefoe we obtain that G fg : LK fg Š is a powe of and thus G fg ˆ LK fg.
Coollay.5. Let p be an odd pime, L a nontivial p-goup and K a nite goup such that K L is a gap goup. Then the following holds. (1) K N is a gap goup fo any nontivial subgoup N of L. () If K fpg < K, then K is a gap goup. Poof. In () we let N be a tivial goup. Let V be a gap K L - module. Regading V L as a K L -module, set W ˆ ResKN KL V L. Then W Kfqg ˆ V KfqgL J V KL fqg ˆ 0 fo any pime q, namely W is L K N -fee. Fo P; H A D K N, it follows that P is a p-goup, PL; HL A D K L and then d W P; H ˆd V PL; HL > 0. Theefoe W is positive on D K N and hence W l dim W 1 V K N is a gap K N -module. 3. Poduct with a symmetic goup. Gap modules fo diect poduct goups 979 Let C n be a cyclic goup of ode n. In this section, by constucting appopiate gap modules, we show that S 5 S 4, S 5 S 5 and S 5 S 4 C ae all gap goups. The poof depends on [5, Theoem 3.5] and the fact that A 4 C is an almost gap goup. Let C G be a complete set of cyclic goups C of G geneated by elements in HnP of -powe ode, fo all P; H A D G. Let C G be a complete set of epesentatives of conjugacy classes of elements C A C G. We denote by G fpg a p-sylow subgoup of G fo a pime p. Poposition 3.1. G ˆ A 4 C is an almost gap goup but not a gap goup. Poof. P G V L G ˆfG f3g g causes that G is not a gap goup. Since G f3g ˆ G fg, the set D G consists of fou elements of type G f3g ; G f3g C. Thus Ind G C R G Ind G C R G Gfg l V G is a equied almost gap G-module, whee R G is the nontivial ieducible C - module. Poposition 3.. The G-module V G is an almost gap G-module fo any nilpotent goup G not of pime powe ode. Poof. Note that G is isomophic to Q p G fpg. Thus if the ode of G is divisible by thee distict pimes, V G is a gap goup by [5, Theoem 0.]. We may assume jgj ˆp a q b fo pimes p and q p > q. Let P; H A D G. Since PG fpg ˆ G implies P ˆ G fpg ˆ G fqg A L G, thee ae no elements P; H A D G such that P B L G. Thus V G is an almost gap G-module by [5, Lemma 0.1]. Poposition 3.3. G ˆ A 5 C is a gap goup.
980 T. Sumi Poof. Let K ˆ A 4 C and W 0 be an almost gap K-module. Set W ˆ Ind G K W 0 and V ˆ W l dim W 1 V G. We show that V is a gap G-module. It su½ces to show that W is positive at all P; H A D G. Note that d W P; H ˆ d W0 K V g 1 Pg; K V g 1 Hg b0: PgK A PnG=K H=P Since K fg is a Sylow -subgoup of G, we have PnG=K H=P 0q. It su½ces to show that K V g 1 Pg B L K. Suppose K V g 1 Pg A L K. Then K V g 1 Pg ˆ K fg. Thus P is a Sylow -subgoup of G but this contacts the existence of H. Hence K V g 1 Pg B L K and W is positive at all P; H A D G. Recalling (.3), given a subgoup L of G, we de ne a G-module V L; G ˆ Ind G L R LŠ R L G, namely an L G -fee G-module emoving non-l G -fee pat 0 p Ind G pg R LŠ R Gf L fom Ind G L R LŠ R. Poposition 3.4. G ˆ S 5 S 4 and S 5 S 5 ae gap goups. Poof. We egad G as a subgoup of S 9. Set K 1 ˆ S 5 A 4, K ˆ A 5 S 4 and K 3 ˆ C 6 S 4, which ae all gap goups. (Also see [5, Lemma 5.6].) We de ne V m ˆ Ind G K m W m fo m ˆ 1; ; 3, whee W m is a gap K m -module. It follows that C G ˆfC ; 1 ; C 4; 1 ; C 1; ; C 1; 4 ; C ; ; C ; 4 ; C 4; ; C 4; 4 ; S ; S 4 ; T ; T 4 g: Hee C i; 1, C 1; i, C i; j, S i and T i ae cyclic subgoups geneated by a i, b i, a i b j, s i and t i espectively i; j ˆ ; 4, whee a ˆ 1; 3, a 4 ˆ 1; ; 3; 4, b ˆ 6; 8, b 4 ˆ 6; 7; 8; 9, s i ˆ a i b4 and t i ˆ a4 b i. Let P; H A D G. If HnP has an element which is conjugate to an element in fa i ; s i j i ˆ ; 4g; esp: fb i ; t i j i ˆ ; 4g; esp: fa ; b i ; a b i j i ˆ ; 4g then V 1 (esp. V, esp. V 3 is positive at P; H. Let L be a subgoup of G of ode 16 geneated by a 4 b, b4, and 6; 7 8; 9. Now assume HnP consists of elements which ae conjugate to elements in fa 4 b i j i ˆ ; 4g. Fo such a pai P; H, thee is an element a of G such that a 1 Ha is a subgoup of L. (Note that G fg ˆ D 8 D 8 has just 4 elements conjugate to g fo each g ˆ a 4 b 4, a 4 b.) Since N G L is a Sylow subgoup G fg,it follows that d V L;G P; H b jn G L j jn G L V a 1 PaLj j Gfg PnG=L H=P j b jn G L =Lj ˆ > 0: Putting all togethe, V L; G l 3 V G l 0 3 iˆ1 V i is a gap G-module.
Gap modules fo diect poduct goups 981 Since S 5 S 5 : GŠ ˆ5 is odd, S 5 S 5 is a gap goup by [5, Lemma 0.3]. Remak 3.5. Conside the following subgoups of G ˆ S 5 S 4 : P ˆ ha4 ; b 4 i, H 4 ˆ ha 4 b 4 ; a 4 b4 3i and H ˆ ha 4 b ; a 4 b b4 i. Then P; H 4 and P; H ae elements of D G. N 4 ˆ N G C 4; 4 ˆha 4 ; b 4 ; a b i of ode 3 has just 4 elements which ae conjugate to a 4 b 4 and no elements conjugate to a 4 b. Thus j H 4 np V N 4 jˆ4 and so H 4 V N 4 ˆ 8. Theefoe if H 4 b C 4; 4, then jn 4 =PC 4; 4 V N 4 jˆjn 4 =H 4 V N 4 jˆ4 and d V C4; 4 P; H 4 b4 ˆ. Similaly since N ˆ N G C 4; ˆha 4 ; b ; b4 ; a i G D 8 C C of ode 3 has only 4 elements conjugate to a 4 b and no elements conjugate to a 4 b 4, it follows that d V C4; P; H b. Then in this estimation we only obtain that V C 4; l V C 4; 4 is nonnegative at P; H 4 and P; H. Howeve j PnG=C 4; j Hj=P jˆ8 in fact and thus d V C4; j P; H j ˆ6 fo j ˆ ; 4. Thus W ˆ V C 4; l V C 4; 4 is positive at P; H j fo j ˆ ; 4, and hence 5 V G l V 1 l V l V 3 l W is a gap G-module. Fo G ˆ S 5 S 4 C, the set C G consists of 8 elements. Let K 1 ˆ S 5 A 4 C, K ˆ A 4 S 4 C and K 3 ˆ C 6 S 4 C be subgoups of G. They ae gap goups by Poposition 3.1 and [5, Theoem 3.5]. Similaly by using thei gap goups, we can pove the next poposition. Poposition 3.6. Let V i i ˆ 1; ; 3 be G-modules induced fom gap K i - modules. Let K 5 be a subgoup of G geneated by 1; ; 3; 4 6; 7, 6; 7 8; 9, 6; 8 7; 9, and 10; 11, viewing natually G ˆ S 5 S 4 C as a subgoup of S 11. Then 3 V 1 l V l V 3 l V G l V K 5 ; G is a gap G-module. Futhemoe, S 5 S 5 C is a gap goup. Consideing the simila agument of the poof of Popositions 3.4 and 3.6, we obtain the following poposition. Poposition 3.7. Let G be a nite goup such that P G V L G ˆq and F a subset of C G. We assume that: (1) Fo any element C of F, thee is a gap goup K such that Ca Ka G. () Thee is an L G -fee G-module W which is positive at any P; H A D G such that H contains an element of C G nf as a subgoup. Then G is a gap goup. Poof. Fo each element C of F, pick up a gap subgoup K C of G which includes C and a gap K C -module W C. Then Ind G K C W C is nonnegative on D G,
98 T. Sumi and is positive at P; H A D G if C V g 1 HnP g 0q fo some g A G. Theefoe W l dim W 1 V G l 0 C A F Ind G K C W C is a gap G-module. Let nb9. Note that K ˆ S n 5 A 5 a S n is a gap goup, since A 5 C is a gap goup. Let F H C G be a set of all elements of ode < k, whee k is a powe of such that k a n < k. Then K contains any element of F up to conjugate in G. W ˆ V h 1; ;...; k i; S n ful lls () in Poposition 3.7. Hence S n is a gap goup which has been aleady shown in []. 4. Fakas lemma and the condition NGC. Thoughout this section, we assume that G is a nite goup not of pime powe ode. We conside the following condition NGC: Thee ae a nonempty subset S H D G and positive integes m P; H fo P; H A S such that 4:1 P; H A S m P; H d V P; H ˆ0 fo any L G -fee ieducible G-module V. We denote by NGC(G) the condition NGC fo a goup G. If G fpg ˆ G fqg, then setting S ˆf G fqg ; G g and m G fqg ; G ˆ1, we obtain (4.1). If P G V L G ˆq, then S must be a subset of D G by existence of V G. We give two examples. Fo a dihedal goup D n of ode n, any L D n - fee ieducible module is zeo at f1g; C. Let P 1 ˆ h 1; 3 ; 4 i, H 1 ˆ h 1; ; 3; 4 i, P ˆ h 1; ; 3 i, and H ˆ h 1; ; 3 ; 1; i be subgoups of S 5,and set S ˆf P 1 ; H 1 ; P ; H g. Then d W P 1 ; H 1 d W P ; H ˆ0 fo any L S 5 - fee ieducible module W. (See [6].) Theefoe D n and S 5 satisfy the condition NGC. Heeafte we show that if G is not a gap goup, G satis es the condition NGC. We wite xby (esp. x > y, ifx i b y i (esp. x i > y i fo any i, whee x ˆ t x 1 ;...; x n Š and y ˆ t y 1 ;...; y n Š. Theoem 4. (The duality theoem cf. [1, p. 48]). with enties in Q, let Fo an n m matix A minimize t cx subject to Axbb; xb0 be a pimal poblem and let maximize t by subject to t Ayac; yb0
Gap modules fo diect poduct goups 983 be a poblem which is called the dual poblem. between the pimal and dual poblems holds. Then the following elationship Dual Optimal Infeasible Unbounded Optimal Possible Impossible Impossible Pimal Infeasible Impossible Possible Possible Unbounded Impossible Possible Impossible The duality theoem is poved by applying a linea pogamming ove Q. A key point of the poof is that the (evised) simplex method is closed ove Q. We omit the detail. Lemma 4.3 (Fakas Lemma). Let A be an n m matix with enties in Q. Fo b A Q n, set A; b ˆfx A Q m j Ax ˆ b; xb0g and Y A; b ˆfy A Q n j t Aya0; t by > 0g: Then eithe A; b o Y A; b is empty but not both. Poof. Fist suppose A; b 0q. If it might holds Y A; b 0q, then t yax ˆ tyb ˆ tby > 0 but the inequalities t yaa t 0 and xb0 implies t yaxa0 which is contadiction. Thus Y A; b is empty. Next suppose A; b ˆq. Conside a pimal poblem minimize subject to which has an infeasible solution. whee z ˆ y 1 y and y ˆ y1 y t 0x A A xb b b ; xb0 Then the dual poblem is maximize t bz subject to t Aza0. This poblem has a solution z ˆ 0 and thus it has an unbounded solution. Theefoe thee exists a solution z such that t bz > 0 and then Y A; b 0q. Let n be a numbe of L G -fee ieducible G-modules and m ˆjD G j. We denote by M m; n; Z the set of m n matices with enties in Z. We say that D is a dimension matix of G, ifd A M m; n; Z is a matix whose i; j - enty is d Vj P i ; H i, whee V j uns ove L G -fee ieducible G-modules and P i ; H i uns ove elements of D G. Fo a subset S of D G, a submatix
984 T. Sumi D 0 A M jsj; n; Z of a dimension matix D of G is called a dimension submatix of G ove S. Set and Z k b0 ˆfx A Z k j xb0g ( Z S G ˆ y ˆ t y 1 ;...; y k Š A Zb0 k t D 0 ya0; ) y i > 0 : i If G is a gap goup, then thee is x A Zb0 n such that Dx > 0. The convese is also tue, since W ˆ Pi x iv i is a gap G-module, whee x i is the i-th enty of x. Poposition 4.4. The followings ae equivalent. (1) G is not a gap goup. () Z D G G 0q. (3) Thee ae a nonempty subset S J D G and positive integes m P; H fo P; H A S such that P P; H A S m P; H d V P; H a0 fo any L G -fee ieducible G-module V. Poof. Let D be a dimension matix of G. Set A ˆ D; EŠ, whee E is the identity matix, and b ˆ t 1;...; 1Š. If thee is x ˆ t x 1 ; x Š A A; b, then Dx 1 x ˆ b and thus Dx 1 b b. Take a positive intege k such that kx 1 A Z m. Then D kx 1 bkbbb. Theefoe A; b 0q implies that G is a gap goup. Clealy if G is a gap goup, then A; b 0q holds. Then by Lemma 4.3, G is a gap goup if and only if Y A; b ˆq, equivalently Z D G G ˆq holds. Theefoe (1) and () ae equivalent. It is clea that (3) implies (). To nish the poof we show that () implies (3). Take z A Z D G G. Set S as a set of P it ; H it 's such that the i t -th enty of z is nonzeo, and let m P it ; H it be the i t -th enty of z. Then P P i t ; H it A S m P i t ; H it d Vj P it ; H it a0 clealy holds. Thus if NGC(G ) holds, then G is not a gap goup. Poposition 4.5. Suppose that thee ae an L G -fee G-module W and a subset T J D G such that d W P; H b0 fo any P; H A D G and d W P; H > 0 fo any P; H A T. Then the followings ae equivalent. (1) G is not a gap goup. () Z D G nt G 0q. (3) Thee is a nonempty subset S J D G nt and integes m P; H > 0 fo P; H A S such that P P; H A S m P; H d V P; H a0 fo any L G -fee ieducible G-module V. Poof. Clealy () implies (1) by Poposition 4.4. Suppose that G is not a gap goup. Let D 1 be a dimension submatix of G ove S and D a dimension submatix of G ove D G ns. Then D ˆ t D 1 ; D Š is a dimension matix of G.
Gap modules fo diect poduct goups 985 Let V j 1aja k be a complete set of L G -fee ieducible G-modules. Set y ˆ t y 1 ;...; y k Š, whee W ˆ Pk jˆ1 y jv j. Since thee is a nonzeo vecto x ˆ t x 1 ; x Šb0 such that t xd ˆ tx 1 D 1 t x D a t 0, we have t x 1 D 1 y t x D ya0. Since D 1 y > 0 and D yb0, we obtain that t x 1 D 1 y ˆ tx D y ˆ 0 and thus x 1 ˆ 0. Theefoe t x D a t 0 fo the nonzeo vecto x b 0 and hence () holds. Coollay 4.6. Let G be a nite goup such that P G V L G ˆq. The goup G is a gap goup if and only if Z D G G ˆq holds. This holds fom the existence of V G. The following poposition can be poven by the same manne of the poof of Poposition 4.5. Recall that P G V L G ˆq implies D G ˆD G. Poposition 4.7. Let G be a nite goup not of pime powe ode. Suppose that thee ae an L G -fee G-module W and a subset T J D G such that d W P; H b0 fo any P; H A D G and d W P; H > 0 fo any P; H A T. Then the followings ae equivalent. (1) G is not an almost gap goup. () Z D G nt G 0q. (3) Thee is a nonempty subset S J D G nt and integes m P; H > 0 fo P; H A S such that P P; H A S m P; H d V P; H a0 fo any L G -fee ieducible G-module V. Theoem 4.8. Let G be a nite goup not of pime powe ode. Then Z S G ˆfy A Z k b0 j t Dy ˆ 0; y 0 0g; whee D is a dimension submatix ove S. and only if NGC(G) holds. In paticula G is not a gap goup if Poof. Since (1) and (3) of Poposition 4.4 ae equivalent, NGC(G ) implies G is not a gap goup. Suppose that G is not a gap goup. We show that NGC(G) holds. Let D be a dimension submatix of G ove D G and set c ˆ t 1;...; 1Š A Q n. Note that R GŠ includes all ieducible G-modules. Since V G is a module emoving non-l G -fee, (ieducible) G-modules fom R GŠ, the G- module V G includes any L G -fee ieducible G-modules. Let a A Zb0 n be a vecto coesponding with V G. Thus we obtain that both Da ˆ 0 and t ba > 0 fo any b A Q n such that bb0 and t cb > 0. Then By Lemma 4.3 we get namely, Y t D; b V Y t D; b 0q: t D; b U t D; b ˆq; fx A Q m j t Dx ˆ b; xb0g U fx A Q m j t Dx ˆ b; xb0g ˆq:
986 T. Sumi Then de ning a map f : Z m b0! Z n by f x ˆtDx, the image of f is a subset of Z n nfgb j bb0; t cb > 0g ˆ Z n nfgb j bb0g U f0g: Taking z A Z D G G by Poposition 4.5, f z a0 holds. On the othe hand, the vecto f z belongs to Z n nfgb j bb0g U f0g. Hence we obtain f z ˆ0. We complete the poof. 5. Poduct with the cyclic goup of ode. The pupose of this section is to pove the following lemma. Lemma 5.1. If K is an almost gap goup, then so is G ˆ K C. Combining [5, Theoem 0.4] and Lemma 5.1, we obtain Theoem 1.. Now we show Lemma 5.1. To apply Poposition 4.5, we de ne a subset T of D G. Let W be an almost gap K-module. Let p 1 : G! K and p : G! C be canonical pojections. Fist, set T 1 ˆ D G nd G. The module V 1 ˆ V G is nonnegative on D G and positive on T 1. Second, set T ˆf P; H A D G j p P ˆp H g. Then V ˆ Ind G K W is nonnegative on D G and positive on T. It is clea that V 1 and V ae L G -fee. Note that V P C is an almost gap goup and paticulaly, nonnegative on D G fo any p-goup P p 0. Thid, set T 3 ˆf P; P C A D G jp A P K nl K g. We show that thee is an L G -fee G-module V 3 such that V 3 is nonnegative on D G and positive on T 3, by dividing two cases. Let P; H A T 3. The st case is one whee jkj is divisible by at least two odd pimes. Take an odd pime q such that q divides jkj and addly if P 0 f1g then P is not a q- goup. Then Ind G C q C V C q C is positive at P; H. Set V 3 ˆ 0 p Ind G C p C V C p C, whee p anges ove all odd pimes which divide jkj. Then V 3 is positive on T 3. The second case is one whee jkj ˆ a p b fo some odd pime p and some intege a; bb1. Set L ˆ K fpg C and V 3 ˆ Ind G L V L. If P K V L K ˆq, then K fpg is not a nomal subgoup of K and thus thee is an element g A G fo any P A P K such that L V g 1 Pg < K fpg. If K fpg is a nomal subgoup of K, then L V P < K fpg fo any P A P K nl K. Theefoe we obtain that d V3 P; H ˆ d V L L V g 1 Pg; L V g 1 Hg PgL A PnG=L H=P ˆ p 1 P gk f pg A p 1 P nk=k f pg dim V L LVg 1Pg > 0: Then V 3 is positive on T 3. Putting all togethe, V ˆ V 1 l V l V 3 is nonnegative on D G and positive on T ˆ T 1 U T U T 3.
Gap modules fo diect poduct goups 987 Let V j 1aja g be all ieducible K-modules such that V j is L K -fee K wheneve 1aja a, V fg K j ˆ 0 but V f pg j 0 0 fo some odd pime p wheneve K a < ja b, and V fg j 0 0 wheneve b < ja g. Then any L G -fee ieducible G-module is one of U j ˆ V j n R 1aja a and W k ˆ V k n R G 1aka b. Hee R (esp. R G is the ieducible tivial (esp. nontivial) C -module. Suppose that G is not an almost gap goup. By Poposition 4.7, thee ae a nonempty subset S J D G nt and a nonzeo vecto x A Z g b0 such that t xda t 0. Hee D ˆ d Uj P; H ; d Wk P; H Š is a dimension submatix of G ove S, whee 1aja a and 1aka b. Fo P; H A S, we obtain that P ˆ H V K, p 1 H > P, d Uj P; H ˆ 1 w jpj Vj p 1 h w R p h ˆ d Vj P; p 1 H ; and d Wk P; H ˆ 1 jpj h A HnP h A HnP w Wk p 1 h w RG p h ˆ d Vk P; p 1 H : Let F ˆ d Uj P; H Š be a submatix of D such that D ˆ F; F; F 0 Š fo some matix F 0. Then t xda t 0 implies that t xfa t 0 and t xfa t 0. Hence t xf ˆ t0 holds. On the othe hand, a map D G nt! D K assigning P; H to P; p 1 H is a bijection. (If jkj is odd, then D G nt and D K ae both empty.) Then F ˆ d Vj P; p 1 H Š is a dimension submatix of K. By Poposition 4.7, K is not an almost gap goup, which is contadiction. Theefoe K C is also an almost gap goup. Coollay 5.. The weath poduct K L is a gap goup fo any nite goup K, if L is a gap goup. Poof. It is clea fom the existence of epimophisms K L! L. 6. Poduct with a dihedal goup. Let D n ˆ ha; b j a ˆ b n ˆ ab ˆ 1i be a dihedal goup of ode n. In this section we study which K D n is a gap goup. If K is a gap goup, then so is K D n. We ae also inteesting in the convese poblem. We set Dp G ˆf P; H A D G jp is a p-goupg fo a pime p and D 1 G ˆ f f1g; C A D G g: Poposition 6.1. Let G be a nite goup not of pime powe ode, p an odd pime and Q a nontivial p-goup. The natual pojection p : G Q! G induces a sujection Z D p GQ G Q!Z Dp G G. Futhemoe, it is a bijection if jgj and p ae copime.
988 T. Sumi Poof. Let P; H A Dp G Q. Note that H V Q ˆ P V Q, G Q fg V Q ˆ Q and p P G Q fg ˆp P p GQ fg ˆ p P G fg fo any pime. Thus p P ; p H A Dp G. Fo a G-module V, it follows that d VnR P; H ˆ 1 jpj h A H np jp V Qj w V p h ˆ jpj x A p H np P w V x ˆd V p P ; p H ; whee R egads as the tivial W-module and w V is the chaacte fo V. Thus the pojection p induces a map Z D p GQ G Q!Z Dp G G. We show that the map is sujective. Set S ˆf AQ; B Q j A; B A Dp G g which is a subset of Dp G Q. Let P; H A S. Then d VnW P; H ˆd V p P ; p H dim W Q fo a G-module V and a Q-module W. If V W is L G Q -fee and W is the tivial ieducible Q-module, then V is L G -fee. Thus a dimension submatix D ˆ d VnW P; H Š ove S coincides with d V p P ; p H ; 0;...; 0Š. Note that d V p P ; p H Š is a dimension submatix ove Dp G. Fo x A Z Dp G G, take y A Z D p GQ G Q whose enty coesponding to P; H A D p G Q is the enty of x coesponding to p P ; p H if P; H A S and zeo othewise. Then the map sends y to x. Theefoe the map is sujective. If jgj is a copime to p, then S ˆ Dp G Q which implies that the map is bijective. We complete the poof. This poposition implies as follows. If P G V L G ˆq, thenz D p G G 0q is equivalent to that thee is a nontivial p-goup Q such that G Q is not a gap goup. Futhemoe, if G Q is a gap goup fo some nontivial p-goup Q, so is G R fo any nontivial p-goup R. It also holds in the case whee p ˆ, by Theoem 1. and Poposition.. Coollay 6.. Let K be a p-goup. The goup G ˆ K D n is not a gap goup. Poof. Since f1g;hai A D p D n, Poposition 6.1 yields the assetion. Poposition 6.3. Let p be a pime and let K 1 and K be nite goups not of pime powe ode. If Z D p K 1 K 1 and Z D p K K ae both nonempty, then Z D p K 1 K K 1 K 0q. Poof. We de ne P; H A Dp K 1 K fo P 1 ; H 1 A Dp K 1 and P ; H A Dp K as follows. Set P ˆ P 1 P, which is a p-goup. Take h j A H j such that h j B P j and h j is an element of -powe ode fo j ˆ 1;, and denote by H a subgoup of K 1 K geneated by P and h ˆ h 1 h. It is clea that P; H A Dp K 1 K. Let S be a subset of Dp K 1 K which is the image of the above assignment and D ˆ d VnW P; H Š a dimension submatix ove S. Since
d VnW P; H ˆ 1 w jpj V p 1 hx w W p hx ˆ 1 jpj x A P p 1 ; p A P w V h 1 p 1 w W h p ˆ 1 w jpj V h 1 p 1 w W h p p 1 A P 1 p A P ˆ d V P 1 ; H 1 d W P ; H ; we have d VnW P; H Š ˆ d V P 1 ; H 1 Š n d W P ; H Š. Recall that d V P 1 ; H 1 Š (esp. d W P ; H Š is a dimension submatix ove Dp K 1 (esp. Dp K. Thus x j A Z D p K j K j jˆ1; implies x 1 n x A Z D p K 1 K K 1 K. Remaking 7 p Dp G ˆD 1 G, similaly as in the poof of Poposition 6.3, we obtain the following poposition. Poposition 6.4. Let K 1 and K be nite goups not of pime powe ode such that Z D K 1 K 1 0qand Z D 1 K K 0q. Then Z D K 1 K K 1 K 0qholds. On the othe hand, the G-module V G gives some estiction: Poposition 6.5. Let G be a nite goup such that f1g < G fpg < G fo some odd pime p. Then d V G is positive on D1 G. In paticula, Z D 1 G G ˆq holds. Example 6.6. Let D 4 ˆ h 1; 3; 4 ; 1; 3 ; 4 i and D 8 ˆ h 1; 3; 4 ; 1; ; 3; 4 i be subgoups of S 4. Then D 4 ; D 8 A Z D S 4 S 4. Thus Z D S 4S 4 S 4 S 4 0q which implies that S 4 S 4 is not a gap goup. Repeating, Q n iˆ1 S 4 is also not a gap goup. Now we pove the main theoem. Gap modules fo diect poduct goups 989 Poof of Theoem 1.1. By Theoem 1., G is a gap goup if so is K. If K is of pime ode, Coollay 6. yields the assetion. Let K be a nite goup not of pime powe ode which is not a gap goup. Then thee is a vecto x A Z D K K. By Poposition., it su½ces to show NGC(G) unde the assumption that n is odd, say n ˆ g 1. Let D ˆ d Vj P; H Š A M s; t; Z be a dimension submatix of K ove D K. We de ne P; H 0 A D G fo P; H A D K as follows. Take an element h A HnP of -powe ode. Let H 0 be a subgoup of G which ae geneated by P and ha. Note that H 0 does not depend on the choice of h. Set F ˆ d V 0 P; H 0 Š A M s; t 0 ; Z, whee t 0 is a numbe of j L G -fee ieducible G-modules. We claim that t Fx ˆ 0, which implies Z D G G 0q and thus G is not a gap goup. Let W 1 (esp. W ) be tivial (esp. nontivial) 1-dimensional D n -module and W k 3aka g be all ieducible -dimensional D n -modules. Let V j 1aja b be all ieducible K-
990 T. Sumi modules such that V j is L K -fee wheneve 1aja a but V j is not wheneve a < ja b. Then an L G -fee ieducible G-module is one of V j n W 1 1aja a, V j n W 1aja a and V j n W k 1aja b; 3aka g. Thus t 0 ˆ a bg. We obtain that d Vj nw 1 P; H 0 ˆ 1 w jpj Vj p 1 h w W1 a ˆd Vj P; H h A H 0 np by (.1), whee p 1 : G! K is a canonical pojection. Similaly, we get d Vj nw P; H 0 ˆ d Vj P; H and d Vj nw k P; H 0 ˆ0. Thus F ˆ D; D; 0Š and then t Fx ˆ 0. We complete the poof. Q Coollay 6.7. Let K be a p-goup, a kˆ1 S 4, o S 5. Q b jˆ1 D n j is not a gap goup fo any bb 0 and any n j b 1. Then G ˆ K Poof. Since K is not a gap goup, Coollay 6. and Theoem 1.1 imply NGC K D n1. Thus the poof is completed applying Theoem 1.1 each step by induction on b. Theoem 6.8. Let n k 1aka a be an intege such that n 1 b n b b n a > 1 and let G ˆ Qa kˆ1 S n k be a diect poduct goup of symmetic goups. Then G is a gap goup if and only if eithe ab1 and n 1 b 6 o ab and n 1 ˆ 5, n b 4. This holds fom Popositions., 3.4, Coollay 6.7 and a esult of Dovemann and Hezog []: A symmetic goup S n is a gap goup fo nb6. Refeences [ 1 ] V. ChvaÂtal, Linea Pogamming, W. H. Feeman and company, 1983. [ ] K. H. Dovemann and M. Hezog, Gap conditions fo epesentations of symmetic goups, J. Pue Appl. Algeba, 119 (1997), 113±137. [ 3 ] E. Laitinen and M. Moimoto, Finite goups with smooth one xed point actions on sphees, Foum Math., 10 (1998), 479±50. [ 4 ] M. Moimoto, Deleting-inseting theoems of xed point manifolds, K-theoy, 15 (1998), 13±3. [ 5 ] M. Moimoto, T. Sumi and M. Yanagihaa, Finite goups possessing gap modules, Contemp. Math., 58 (000), 39±34. [ 6 ] M. Moimoto and M. Yanagihaa, The gap condition fo S 5 and GAP pogams, Jou. Fac. Env. Sci. Tech., Okayama Univ., 1 (1996), 1±13. [ 7 ] R. Olive, Fixed point sets of goup actions on nite acyclic complexes, Comment. Math. Helv., 50 (1975), 155±177. Toshio Sumi Depatment of At and Infomation Design Faculty of Design Kyushu Institute of Design Shiobau 4-9-1 Fukuoka, 815-8540, Japan E-mail: sumi@kyushu-id.ac.jp