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4.2 cor 5

# 4.2 cor 5 - GROUPS SUPPLEMENTARY TOPICS Theorem(see...

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GROUPS: SUPPLEMENTARY TOPICS Theorem (see Proposition 2.59, page 172): If G is a finite abelian group, then G has a subgroup of order d for every divisor d of | G | . Theorem (Cayley) (see Theorem 2.66, page 178): Every group G is isomorphic to a subgroup of the symmetric group S G . In particular, if | G | = n, then G is isomorphic to a subgroup of S n . Proof : For each a G, define τ a : G -→ G by τ a ( x ) = ax for every x G (note that if a = 1 , then τ a is not a homomorphism, since τ a (1) = a · 1 = a = 1). We show that τ a is a permutation (bijection) of elements from G. In fact, first note that for any a, b G we have ( τ a τ b )( x ) = τ a ( τ b ( x )) = τ a ( bx ) = a ( bx ) = ( ab ) x = τ ab ( x ) , so τ a τ b = τ ab . It follows that each τ a is a bijection, since its inverse is τ a - 1 : τ a τ a - 1 = τ aa - 1 = τ 1 = 1 G = τ a - 1 a , and so τ a S G , i.e. τ is a permutation. Define ϕ : G -→ S G by ϕ ( a ) = τ a . Rewriting, we get ϕ ( a ) ϕ ( b ) = τ a τ b = τ ab = ϕ ( ab ) , so that ϕ is a homomorphism. Finally, ϕ is an injection. If ϕ ( a ) = ϕ ( b ) , then τ a = τ b , which means τ a ( x ) = τ b ( x ) for all x G ; in particular, when x = 1 , this gives a = b, as desired.

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