Name: MAT 444 Test 2 Instructor: Helene Barcelo March 1, 2004

Transcription

1 MAT Test 2 Istructor: Helee Barcelo March, 200 Name: You ca take up to 2 hours for completig this exam. Close book, otes ad calculator. Do ot use your ow scratch paper. Write each solutio i the space provided, ot o scratch paper. If you eed more room, write o the back of the page. If you still eed more room, ask for scratch paper. Show your reasoig o all problems; do ot simply write a aswer. Your solutios must be complete ad orgaized, otherwise poits may be deducted. Do all 6 problems. There are a total of 75 poits.

2 Test 2 Page 2. (0 poits) True-False. Give a brief reaso (o detailed proofs) for each aswer. (a) Let S be a set o which a group G operates, ad let H = { g G gs = s, s S}. H is a ormal subgroup of G. TRUE It is easy to see that H is a subgroup. Let g G, h H the ( ghg ( s)) = ghg ( ( ( s))) = ghg ( ( ( s)) = gg ( ( s)) = s. So, ad ghg H g G h H. Thus, H G. (b) If H is a fiite subgroup of the group of symmetries of a lattice C, D where =,,6. FALSE = 5 is ot a possibility. See textbook (p. 69). 2 L(i ) the H is oe of (c) The dihedral group D is geerated by 2 elemets x, y which satisfy the relatios 2 x =, y =, ad yx = xy FALSE 2 D = x, y ; x, y, yxyx i.e.: yx x y ad ot xy =. (d) The poit group of a discrete group G is cyclic or dihedral. TRUE The poit group of a discrete group is discrete ad a subgroup of O. But all discrete subgroups of O are either C or D for some. (e) Let S be a G-set, ad let s S. The order of G is equal to the product of the order of the stabilizer of s by the order of the orbit of s. TRUE Coutig formula Typeset by Georgea Loretz

3 Test 2 Page 3 2. (0 poits) Let G be a fiite subgroup of the group O of rigid motios which fix the origi, ad assume that G cotais oly rotatios. Prove that G is a cyclic group. See textbook, p. 65. Typeset by Georgea Loretz

4 Test 2 Page 3. (5 poits) What is the stabilizer of the coset ah for the operatio (left multiplicatio) of G o / G H? Stab( ah ) = aha, sice Stab( ah ) = { g G gah = ah} but gah = ah h H s.t. ga = ah g = aha = aha Typeset by Georgea Loretz

5 Test 2 Page 5. Let G be a group, let S be a G-set, ad deote by Perm(S) the group of its permutatios. Give a actio of G o S, we ca defie φ : G Perm( S) by the rule φ ( g) = mg, where m g is left multiplicatio by g, (that is, the way g acts o S). (a) (0 poits) Prove that φ is a homomorphism. φ ( ) = but gg 2 m gg 2 φ( gg) = φ( g) φ( g) 2 2 m () s = ( g g )() s gg 2 2 = g ( g ( s)) 2 = g ( m ( s)) g2 = m ( m ( s)) g g2 = m m )( s) s S g g2 (b) (0 poits) Prove that this establishes a bijective correspodece betwee the set of all actios of G o S ad the set of all homomorphisms from G to Perm(S). From (a) we have a correspodece betwee a actio of G ad φ : G Perm( S). It remais to show that this correspodece is a bijectio. Clearly, it is oe-to-oe. We are left to show that it is ito. Give ay homomorphism φ : G Perm( S) we ca defie a actio of G o S by gs ( ) = φ( g)( s). We must show this is a actio: ( s) = φ()( s) = s s S sice φ () = idetify permutatio, (φ is a homomorphism) ad ( gg 2) s= φ( gg 2)( s) = ( φ( g) φ( g2)( s) = φ( g)( φ( g2)( s)) = g( g2( s)). Typeset by Georgea Loretz

6 Test 2 Page 6 5. (0 poits) a group G acts faithfully o a set S of five elemets, ad there are two orbits, oe of order 3 ad oe of order 2. what are the possibilities for G? Let S = {,2,3,,5} ad O = {, 2, 3} O = {,, 5}. Let S{, 2,3} S3 be the set of all permutatios of O, ad let S{,5} S2 be the set of all permutatios of O. Sice G operates o O ad O separately, there is a homomorphism φ : G S3 S2 give by φ = where 2 ( g) ( mg, mg ) m g represet the actio of G o O ad 2 m g represet the actio of g o O. (See exercise 5.8 #). Sice G acts faithfully o S, φ is oe-to-oe, ad thus G φ( G). The oly subgroup of S {,5} that acts trasitively o {, 5} is S{,5} S2 itself. O the other had, there are 2 subgroups of S 3 that ca act trasitively o {, 2, 3}: S 3 itself ad C3 (23) = {, (23), (32)}. Thus, the possibilities for G are G S S or G C S Typeset by Georgea Loretz

7 Test 2 Page 7 6. (a) (5 poits) Prove that the set Aut G of automorphisms (bijective homomorphisms of G to G) of a group G forms a group. Let Aut G = { f : G G / f is a bijective homomorphism} with law of compositio beig compositio of fuctios which is associative. i) Let i : G G be the idetify map, i( g) = g, g G. Clearly i is a automorphism ad f Aut G, f i = i f = f. ii) If f, Aut G the compositio f is certaily a bijectio. Is it a homomorphism? ( f )( gh) = f (( gh)) = f (( g) ()) h = f (( g)) f(()) h sice f, are homomorphisms. But f ( ( g)) f( ( h)) = ( f ) ( g) ( f ) ( h) f is a homomorphism. iii) If f Aut G, the f exists ad is a bijectio ad it is easy to show that f is also a homomorphism. The proof is similar to that i (ii). (b) (0 poits) Prove that the map ψ : G AutG defied by g (cojugatio by g) is a homomorphism, ad determie its kerel. Let ψ : G AutG be ψ ( g) = g where g : G G ad g ( h) = ghg, h, g G. ψ ( gg ) = gg which is give by gg ( h) = gg h( gg ) = gghg ( ) g = gg ( hg ) = gg ( ( h)) gg = g g = ψ( g) ψ( g ); hece ψ is a homomorphism. Ker ψ = { g G g = i} = { g G g ( h) = h, h G} = { g G ghg = h h G} = ( G) the ceter of G. (c) (5 poits) The automorphisms which are cojugatio by a group elemet are called ier automorphisms. Prove that the set of ier automorphisms, the image of ψ, is a ormal subgroup of Aut G. Sice ψ is a homomorphism of group, the the image of ψ is a subgroup of Aut G. It remais to show that it is a ormal subgroup ψ ( G) = { g / g G}. Let f Aut G, the f exists ad belogs to Aut G. ( f gf )( h) fg( f ( h)) f( gf ( h) g = = ) = f ( g) h f( g ), h G. Sice f Aut G f ( g) G ad f( g ) = f ( g). Thus, f gf is cojugatio by a elemet, f ( g), of G. Typeset by Georgea Loretz