Isomorphisms Consider the following Cayley tables for the groups Z 4, U(), R (= the group of symmetries of a nonsquare rhombus, consisting of four elements: the two rotations about the center, R 8, and the reflections and about the vertical and horizontal axes): Z 4 U() R R 8 R 8 R 8 R 8 R 8 R 8 Notice the similarities between the Cayley tables for Z 4 and U(). Indeed, Z 4 = and U()=, so the structures of these two groups are identical in the sense that the mapping ϕ:z 4 U() given by a, a, a, a turns the Cayley table of Z 4 into the Cayley table of U(); in particular, the bijection ϕ has the key property that for any pair of elements a,b Z 4, ϕ(a +b (mod4)) =ϕ(a) ϕ (b) (mod). That is, ϕ is operation-preserving.
The third group, R, is noticeably different from the other two. Despite the fact that it has the same order as the other two groups, its Cayley table indicates that its structure is different: R has no elements of order 4, while Z 4 and U() do. This means that no bijective function can be defined between R and either of Z 4 or U() which respects the operations of both groups. This discussion leads to the following definitions. A bijection ϕ:g G (a function that is one-to-one and onto) between a pair of groups G and G is called an isomorphism if for all a,b G, we have ϕ(a b) =ϕ(a)oϕ(b), where is the operation in G and o is the operation in G. Any two groups G and G are isomorphic to each other if there exists some isomorphism ϕ:g G between them. When this is the case, we write G G. Isomorphisms reveal the fact that a pair of groups are structurally the same. For instance, (R, +), the group of real numbers under addition, is isomorphic to (R +, ), the group of positive real numbers under multiplication, via the isomorphism r a e r ; the group of powers of under multiplication is isomorphic to Z, the group of integers under addition, via the isomorphism n a n.
Theorem [Cayley, 854] Any group is isomorphic to a group of permutations. Proof Let G be a group. For every element g G, define a function T g :G G by the formula T g (x) = gx (the notation is chosen to suggest that T g is translation of the group G by the element g). Each function T g is one-to-one since T g (x) = T g (y) gx = gy x = y; further, T g is onto since for any y G, T g (g y) = y. Since T g is a bijection, it permutes the elements of G. Now let G = {T g g G}. We show that this is a group under function composition: () closure follows from the fact that (T g T h )(x) = T g (T h (x)) = T g (hx) = g (hx) = (gh)x = T gh (x); () associativity holds because function composition is always associative; () the identity element is the identity function T e (x) = ex = x; and (4) the inverse of the element T g is T g, since T g T g (x) = T g (g x) = gg x = x = T e (x) = T g T g (x). The function ϕ:g G given by ϕ(g ) = T g is the isomorphism that proves the theorem, because () ϕ(g ) =ϕ(h) T g = T h T g (e) = T h (e) ge = he g = h
shows that ϕ is one-to-one; () ϕ is onto by the way that G was constructed; and () ϕ is operationpreserving because ϕ(gh) = T gh = T g T h =ϕ(g)ϕ(h). // The group G constructed in the proof of Cayley s Theorem is called the left regular representation of G (the right regular representation is the similarly defined group of permutations of the form U g (x) = xg ). It gives us a way to make concrete the operation of an abstract group G, and it also points out that all groups are permutation groups. Theorem If ϕ:g G is a group isomorphism, then. ϕ takes the identity of G to the identity of G ;. for all x G and integers n, ϕ(x n ) =ϕ(x) n ;. for all x, y G, xy = yx ϕ(x) =ϕ(y); 4. G = x G = ϕ(x) ; 5. ϕ(x) = x ; 6. for any group element a G and integer k, the number of solutions of the equation X k = a in G is the same as the number of solutions to the equation X k =ϕ(a) in G ;. G is Abelian G is Abelian; 8. G is cyclic G is cyclic;. ϕ is an isomorphism from G to G;. if is a subgroup of G, then ϕ( ) is a subgroup of G. Proof Omitted. //
Automorphisms It may be that G and G are the same group identically: an isomorphism ϕ:g G is called an automorphism of G. (An automorphism need not be the identity map. For instance, the map ϕ:c C on the group of complex numbers under addition given by a +bi a a bi is an automorphism also known as complex conjugation. While an isomorphism can reveal that groups G and G with different definitions are in fact the same group structurally, an automorphism of the group G can reaveal the structure within the group G. Given a group G and an element a G, the map ϕ a :G G given by ϕ a (x) = axa is an automorphism (why?), called the inner automorphism induced by a. Theorem The set Aut(G) of automorphisms of the group G is itself a group under composition. Further, the set Inn(G) of inner automorphisms of G is a subgroup of Aut(G). Proof If ϕ:g G and ψ:g G are bijections, then so is ϕψ. Also,
(ϕψ )(xy) =ϕ(ψ (xy)) =ϕ(ψ (x) ψ (y)) =ϕ(ψ (x)) ϕ (ψ (y)) = (ϕψ )(x) (ϕψ )(y) shows that ϕψ is operation-preserving. That is, Aut(G) is closed under composition. Associativity in Aut(G) follows from the fact that composition is always associative. The identity map on G is clearly the identity element in Aut(G). Finally, if ϕ Aut(G), then we can use the fact that ϕ is a bijection to find, for any elements x, y G, the unique elements a =ϕ (x) and b =ϕ (y). With these labels we see that not only is ϕ a bijection on G, but ϕ(ab) =ϕ(a)ϕ(b) = xy ϕ (xy) = ab =ϕ (x)ϕ (y), which shows that ϕ is operation-preserving, i.e., ϕ Aut(G). Thus, Aut(G) is a group. To show that Inn(G) is a subgroup of Aut(G), note that composing the inner automorphisms ϕ a and ϕ b produces the inner automorphism ϕ ab, because
ϕ a ϕ b (x) =ϕ a (bxb ) = a(bxb )a = (ab)x (b a ) = (ab)x (ab) =ϕ ab (x) That is, Inn(G) is closed under composition. Further, the computations ϕ a ϕ a (x) =ϕ a (a xa) = a(a xa)a = (aa )x(aa ) = x ϕ a ϕ a (x) =ϕ a (axa ) = a (axa )a = (a a)x(a a) = x show that ϕ a =ϕ a Inn(G). Thus Inn(G) is a subgroup of Aut(G). // An important result in number theory is to determine the structure of Aut(Z n ). Theorem If n is a positive integer, Aut(Z n ) U(n). Proof If α Aut(Z n ), then for any k Z n, α(k) =α( + 4 +L+ 4 ) =α () 44 +α 4 () 4 +Lα() 44 = kα (). k Thus, the images under the automorphism α are completely determined by the single value α(). In particular, α() must be an element of U(n). For if it is not, then there is some non-zero value of k Z n for which kα() (modn), from which it follows k
that α(k) α() (modn), contradicting that α is an automorphism. Therefore, the map α aα() is a function from Aut(Z n ) to U(n). This map is one-to-one since if α,β Aut(Z n ) and α() = β(), then from our observation above, α(k) = kα() = kβ() = β(k), so α = β, that is, the map is one-to-one. This map is also onto, for if u U(n), then the function from Z n to Z n given by α(k) = uk (modn) is easily seen to be an automorphism (details left to an exercise) and α() = u. This shows that α aα() is a bijection between Aut(Z n ) and U(n). It is also operation-preserving, for if α,β Aut(Z n ), then αβ a (αβ)() =α(β()) =α ()β(). Thus, we have shown that Aut(Z n ) and U(n) are isomorphic. //