# Solutions to Problem Set 19

Math 345/504

4-2-2018

1. Let H be the subset of consisting of all matrices of the form

(a) Prove that H is a subgroup of .

(b) Prove that H is normal.

(a) If , then

Thus, H is closed under multiplication.

: The identity is in H.

If , then

Thus, H is closed under taking inverses.

Hence, H is a subgroup of .

(b) Let and . Then

Therefore, H is normal.

2. Use the Universal Property of the Quotient to show that thefunction defined by induces a function defined by

First,

Hence, f is a group map.

Next, if , then

By the Universal Property of the Quotient, f induces a function defined by

3. Show that the function defined by does not induce a function defined by

Specifically, show that f is a group map, but is not.

First,

Hence, f is a group map.

However,

Thus, , so is not a group map.

What lies in our power to do, it lies in our power not to do. - Aristotle

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