# Math homework solutions

HOMEWORK 3

Due Thursday, April 27, at the beginning of discussion

1. (moved to next homework) Write the Cayley table for the Dihedral group D8 with 8

elements. Why is D8 not isomorphic to Z/8Z?

2. Prove or disprove: If G is a group and g, h 2 G, then (gh)1 = h1g1.

3. Prove or disprove: If G is a group, (g1)1 = g.

4. Which of the following set-operation pairs are groups? All matrices are with real entries.

Please prove each answer but feel free to cite any facts from linear algebra.

(a) n ⇥ n matrices with matrix addition.

(b) n ⇥ n matrices with matrix multiplication.

(c) n ⇥ n diagonal matrices with matrix multiplication.

(d) n ⇥ n diagonal matrices with no zero-entries in the diagonal, with matrix multiplication.

(e) n ⇥ n matrices with non-zero determinant (this is called GLn(R)), with matrix

multiplication.

(f) n ⇥ n matrices with determinant +1, with matrix multiplication.

(g) n ⇥ n orthogonal matrices (an orthogonal matrix is a matrix whose columns are

ortho-normal to each other, equivalently AT A = I), with matrix multiplication.

5. Prove that the intersection of two subgroups is always a subgroup. i.e. If G is a group

and H and K are subgroups of G, then H \ K is also a subgroup of G.

6. Give an example of two groups with 9 elements each which are not isomorphic to each

other (and prove it).

7. Assume that G is a group such that for all x 2 G, x ⇤ x = e. Prove that G is an abelian

group.

8. Let (G, ?) be a group. Define the center of G by

Z(G) := {x 2 G: x ? a = a ? x, 8a 2 G}.

The set Z(G) consists of all elements of G that commute with every possible element

of the group. For example, one can say that the matrix 4I belongs to the center of

(GL(2, R), ·) because (4I)A = A(4I) for all A in GL(2, R), since both sides are equal to

4A.

(a) Show that, for every group G, the center Z(G) is a subgroup of G.

(b) Find the center of (Z4, +) and (this part is moved to next homework) the center of

D6, the dihedral group. (You should be able to tell from the group table.)

(c) One could say that “the center Z(G) measures the abelian-ness of a group G”.

Please interpret this statement.1

1Recall that a group (G, ?) is called abelian if the operation ? is commutative. Hint: What is Z(G) equal

to when G is abelian?

1