189-235A: Basic Algebra I Assignment 10

Due: Wednesday, November 24

1. Show that the conjugate elements a and bab^{−1} in a group G have the same order.

2. Show that the intersection of two subgroups H_{1} and H_{2} of a group G is a subgroup of G. What about unions of subgroups?

3. If a is an element of a finite group G of cardinality n, show that a^{n} = 1. Apply this general fact to the group G = Z_{p}^{×} (under multiplication) to give another proof of Fermat’s Little Theorem that p divides a^{p} −a for all integers a when p is prime.

4. Let S be a subset of a group G. The centraliser of S, denoted Z(S), is the set of a ∈ G which commute with every s ∈ S, i.e., such that as = sa for all s ∈ S. Show that Z(S) is a subgroup of G.

5. Let G_{1} be the group of strictly positive real numbers, under multiplication, and let G_{2} be the group of all real numbers, under addition. Show that G_{1} and G_{2} are isomorphic.

6. Let f : G_{1} ⟶ G_{2} be a homomorphism of groups, and let a be an element of G_{1} of finite order. Show that the order of f(a) in G_{2} divides the order of a in G_{1}, and that these two orders are equal if f is injective.

7. Recall the ring H = {a + bi + cj + dk, a, b, c, d ∈ R} of quaternions defined by the multiplication rules

i^{2} = j^{2} = k^{2} = −1, ij = −ji = k, ki = −ik = j, jk = −kj = i.

Show that the subset G = {1, −1, i, −i, j, −j, k, −k} is a subgroup of the multiplicative group of non-zero elements of H. Show that G is not isomorphic to the dihedral group D_{8} of order 8.

Extra Credit

9. Show that the groups GL_{2}(Z_{2}) and S_{3} are isomorphic, by writing down a specific isomorphism. (Hint: realize each matrix as a permutation on the non-zero column vectors with entries in Z_{2}).

10. The conjugacy class of a in a group G is the set of all elements of G which are of the form gag^{−1} for some g ∈ G. Show that a normal subgroup of G is a disjoint union of conjugacy classes. List the conjugacy classes in S_{4} and use this to give a complete list of all the normal subgroups of S_{4}. Same question for S_{5}.