Solutions to Problem Set 4

Math 345/504

2-2-2018

1. In each case, a group and a subset of the group are given. Check EACH axiom for a subgroup. If the axiom holds, prove it; if the axiom doesn't hold, give a specific counterexample.

(a) In the group of invertible matrices with real entries under matrix multiplication, consider the subset

(b) In the group of positive real numbers under multiplication, consider the subset

(c) In the group of positive real numbers under multiplication, consider the subset of positive integers

(d) In the group of pairs of real numbers under componentwise (vector) addition, consider the subset

(So, for example, and are elements of P.)

(a) Let , where .

Hence, H is closed under multiplication.

Let , where .

Hence, H is closed under taking inverses. Thus, H is a subgroup.

(b) Let .

Hence, K is closed under multiplication.

Let .

Hence, H is closed under taking inverses. Thus, H is a subgroup.

(c) If , then m and n are positive integers, so is a positive integer, and . Thus, L is closed under multiplication.

, so L contains the identity.

However, , but . Hence, L is not a subgroup.

(d) , so P contains the identity.

If , then , and if , then . Thus, P is closed under taking inverses.

However, and , but

Thus, P is not closed under addition. Hence, P is not a subgroup.

2. (a) Define by . Prove that f is a group map.

(b) Define by . Prove that g is not a group map.

(a) Let . Then

Therefore, f is a group map.

(b)

Thus, , so g is not a group map.

3. Let G be a group, and suppose that for all . Prove that G is abelian.

Let . I must show that . Since for all , it follows that

Hence,

Therefore, G is abelian.

[MATH 504]

4. Let G be a group. Define by

Prove that f is a group map if and only if G is abelian.

Note: Be sure that you prove both implications.

Let . Suppose f is a group map. Then

Hence, G is abelian.

Suppose G is abelian. Then

Hence, f is a group map.

You probably wouldn't worry about what people think of you if you could know how seldom they do. - Olin Miller

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