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^{-1}y^{-1} (where in each case multiplication and inverses are done using the original operation in G).

Prove or disprove: For any group G, G

^{*}is a semigroup.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.