Here are the /Solutions.

1. The odd get even

Let G be a subgroup of S_n. Show that if G contains an odd permutation, then |G| is even.

2. Damaging a group

Let G be a group. Consider the algebra G^* obtained by replacing the multiplication operation in G with x*y = xy^-1 and G^** obtained by replacing the multiplication operator in G with x**y = x^-1y^-1 (where in each case multiplication and inverses are done using the original operation in G).

1. Prove or disprove: For any group G, G^* is a semigroup.

2. Prove or disprove: For any group G, G^** is a semigroup.

3. Whirling polygons

Show that D_n has a subgroup of size m if and only if m divides 2n.

4. Cocosets

Given a group G with subgroups H and K, define HK = { hk | h ∈ H, k ∈ K }. Show that HK is a subgroup of G if and only if HK = KH.

5. Rational quotients

Let ℚ be the additive group of the rationals, i.e. the group whose elements are numbers of the form n/m for integers n and m ≠ 0 and whose operation is the usual addition operation for fractions, and let ℤ be the additive group of the integers, which we will treat as equal to the subgroup of the rationals generated by 1 = 1/1. Prove or disprove: ℤ is isomorphic to a subgroup of ℚ/ℤ.