Mathematics 243, section 3 – Algebraic Structures Problem Set 8 due: Friday, November 16

‘A’ Section

1. Consider the 2 × 2 matrices I_{2}, S, X, Y, D, T defined by

I_{2} = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , S = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix})
X = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) , Y = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix})
D = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , T = (\begin{matrix}0 & −1 \\ −1 & 0 \end{matrix})

Let G = {I_{2}, S, S^{2}, S^{3}, X, Y, D, T }. Construct the operation table for matrix multiplication on this set and verify that G is a group under this operation.

2. a. Let G = ℤ_{12} under the operation of addition modulo 12. Determine the cyclic subgroups ⟨[a]⟩ for all [a] ∈ ℤ_{12}.

b. Conjecture a general formula for the number of elements in ⟨[a]⟩ in terms of the integers a and 12. Check out your conjecture on the corresponding list of cyclic subgroups of ℤ_{20} constructed in class on November 14.

‘B’ Section

1. Let

G = \{(\begin{matrix}0 & a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{matrix}) | a, b, c ∈ ℝ\}

Show that G is a group under the operation of matrix addition.

2. Let

G = \{(\begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}) | a, b, c ∈ ℝ\}

Is G is a group under the operation of matrix multiplication. If so, say why; if not say which properties in the definition fail.

3. Let G be a group, and consider the relation R on G defined by xRy ↔ y = axa^{−1} for some a ∈ G. (If xRy is true, the a that works in the equation will depend on x, y.) R is called the conjugacy relation on G. a. Show that conjugacy is an equivalence relation on G.

b. Show that G is an abelian group if and only if the equivalence classes for the conjugacy relation satisfy [x] = {x} for all x ∈ G.

c. More generally, show that if [x] = {x} for some element x ∈ G, then xa = ax for all a ∈ G and conversely. The set of such elements x is called the center of G: The center is the subset of G defined by:

Z(G) = {x ∈ G | xa = ax for all a in G}.

d. Show that the center of G (as defined in part c) is a subgroup of G.

e. Let x ∈ G be a fixed element and define C_{x} = {a ∈ G | x = axa^{−1}}. Show that C_{x} is a subgroup of G (C_{x} is called the centralizer of x).

f. Let G = S(A) be the group of permutations of A = {a, b, c}. Using the names for the elements of this group we introduced in Problem Set 3, find all of the equivalence classes for the conjugacy relation on G (there are three of them), determine the centralizer of each element of G, and the center of G. How are the sizes of the equivalence class of x and the number of elements of the centralizer of x related in each case?

4. Let H and K be subgroups of a group G.

a. Show that H ∩ K is also subgroup of G.

b. Find an example where H ∪K is a subgroup of G and one where H ∪K is not a subgroup of G.