Re: [代數] subgroup of a finite group
※ 引述《wsx02 ()》之銘言：
: If H is a subgroup of a finite group G, then for any a,b in G
: |aH| = |H|
: 請問這個要怎麼證?
: 謝謝
For every element h of H, define a map f:H->aH by f(h)=ah. Then check f is a
bijective map:
(1-1) Let h and h' be elements of H. Assume that f(h)=f(h'). Then we have
-1 -1 -1 -1
h=a (ah)=a f(h)=a f(h')=a (ah')=h', so f is injective.
(onto) Let x be an element of aH. By the definition of aH, there is an element,
say y, of H such that x=ay. Then we have f(y)=ay=x, and so f is surjective.
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