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May 25th, 2017, 07:59 AM   #1
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Suppose G is a group, a ∈ G and H is a subgroup of G.

1. Show that aH and H have the same cardinality by exhibiting a bijection between the two sets. (Similarly, one can show that Ha and H have the same cardinality, so aH and Ha always have the same cardinality.)

I've never actually demonstrated bijections between cosets and subgroups before, how would I go about this?

2. Show, by means of an example, that aH is not necessarily equal to Ha. (Hint: it has to be a non-commutative group.)

Would any non-commutative group work for this? Apologies, I can't get my intuition around for this problem
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