# Solutions to Problem Set 18

Math 345/504

3-28-2018

1. In the group , the cosets of the subgroup are

(a) Compute

(b) Compute

(c) Compute

(d) Make a table showing the order of each of the 4 cosets in the quotient group .

Use your table to determine whether is isomorphic to or to .

(a)

(b)

(c) Compute

(d)

Since there are elements of order 4, the quotient group is isomorphic to .

2. The group is

The cosets of the subgroup are

(a) Compute

(b) Compute

(c) Compute

(d) Make a table showing the order of each of the 4 cosets in the quotient group .

Use your table to determine whether is isomorphic to or to .

(a)

(b)

(c)

(d)

Since every non-identity element has order 2, the quotient group is .

3. is the group of real matrices under matrix multiplication. Let

Prove that H is not a normal subgroup of by specific counterexample.

[Math 504]

4. Two of the sporadic simple groups are known as the Monster Group and the Baby Monster Group. Find out their orders, and give a brief (one-sentence) description of the Monster. (That is, describe in general terms how it was constructed.)

The Monster is a sporadic simple group of order

It was constructed in 1982 by Robert Griess, and is a group of rotations in 196883-dimensional space.

The Baby Monster is a sporadic simple group of order

You are all you will ever have for certain. - June Havoc

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