SlOMTH 3175 Group Theory (Prof.Todorov) Quiz 3, Solutions Name:
1. Let G be a group and let H be a subgroup of G. Let a E G'. Prove that the set
aHa
(b)
(C)
H1 = cfw_aha1 I where h E H is a subgroup of G. Proof:
Claim 1: (LI-Ia.1 C G, i.e. is a subset of

MTH 3175 Group Theory Spring 2010 - Prof. Iarrobino and Prof. Todorov
Solutions - AI
Some solutions are reworded/rewritten in order to be more in the style that I was teaching
(with GT in front of it).
FINAL EXAM
Problems #1-8 will each count ten points.

W
SllMTH 3175 Group Theory (Prof.Todorov)
1. Consider external direct product: Z5 63 Z15.
(a) What is the order of (2, 3) E 26 6 215?
Answer:
0 Order of (51,15) is f(a,b)| 2 lcm(|a|, |b|).
o ](2,3)| = lcm([2|, J3[) = lcm(3,5) = 15
since [2] = 6/2 2 3 in

cfw_1
SllMTH 3175 Group Theory (Prof.Todorov) Quiz 3, Solutions Name:
1. Let G be a group and let H be a subgroup of G. Let a E G. Prove that the set
(JLHa1 = cfw_aha~1 I where h e H is a. subgroup of G.
Proof:
(0)
(C)
Claim 1: aHa,1 C G, i.e. is a subse

MTH 3175 Group Theory Spring 2010 - Prof. Iarrobino and Prof. Todorov
Name.
FINAL EXAM
Problems #1-8 will each count ten points. The best 3 of # 9-14 will count ten points each.
(*) denotes extra credit. There are several formulas at the end. Good luck!
1

SllMTH 3l75 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:
Dihedral group D4
1. Let D4 =< p,t | p4 = e, t2 = e, tpt = p]L > be the dihedral group.
(a) Write the Cayley table for D4. You may use the fact that cfw_e, p, p2,p3,t, tp, tpz, tp3
(b

FllMTH 3175 Group Theory (Prof.Todorov) Quiz 3, Solutions Name:
1. (a) Find the conjugate of (1234) (56) by a = (25) in 87.
Denition: A conjugate of a by a is (158(0) = aaal.
Remark: If order of an element a in a group is |a] = n, then (11 = tin1.
If a =

MTH 3175 Group Theory Spring 2011 FE - Prof. Iarrobino and Prof. Todorov
1
2
3
4
5
6
7
8
9
10
11
12
13
total
Name: Solutions - AI
FINAL EXAM
The first 7 problems will each count 10 points. The best 3 of # 8-13 will count 10 points each.
Total is 100 point

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SlOMTH 3175 Group Theory (Prof.T0dorov) Quiz 4 (Practice and Some solutions) Name:
1. Let D, be the dihedral group given by the generators and relations as
D4=<t,p[t2=e,p4=e,tpt=p1>.
(a) Prove that cfw_e, p2, t, tp2 is a subgroup of D4.
Proof: Let S