# math algebra (master )

**Question description**

1) List all subgroups of S_3.

2) In S_5, how many elements x are there such that x^5=identity?

3) Let H be a subgroup of a group G. Find a bijection between the set of left cosets with respect to H and the set of right cosets with respect to H.

4) Let H be a subgroup of a group G. If [G:H]=2, prove that H is a normal subgroup of G.

5) Let f be an homomorphism from the group of integers Z to itself. Show that f is completely determined by its action on 1: If f(1) = r, then f is multiplication by r; in other words, f(n) = rn for every integer n.

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