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Here are the /Solutions.

1. The odd get even

Let G be a subgroup of Sn. 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 Dn 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.

2014-06-17 11:57