F13MTH 3175 Group Theory (Prof.Todorov) Final (Practice)
1
3 PERMUTATIONS
Some Denitions
For your convenience, we recall some of the denitions:
A group G is called simple if it has no proper normal subgroups, i.e. the only normal subgroups
are cfw_e and
F13MTH 3175 Group Theory (Prof.Todorov)
Quiz 4 (Practice)
Name:
Some problems are really easy, some are harder, some are repetitions.
1. Let Sn be the group of permutations on n elements cfw_1, 2, 3, . . . , n. Let An be the subgroup
of even permutations.
MTH 3175 Group Theory (Prof.Todorov)
Quiz 2 (Practice)
Name:
Please explain all your work ! When using theorems, write their statements.
1. Let G be a group and let H and K be subgroups of G. Prove that H K is a subgroup of G.
2. Let G = U (20). Is G cycl
S10MTH 3175 Group Theory (Prof.Todorov)
Quiz 5 (Practice)
Name:
1. Let Sn be the group of permutations on n elements cfw_1, 2, 3, . . . , n. Let An be the subgroup
of even permutations. Prove that An is normal subgroup of Sn .
2. Let S15 be the group of p
S11MTH 3175 Group Theory (Prof.Todorov)
Quiz 2 Solutions
Name:
Please explain all your work ! When using theorems, write their statements.
1. Let G be a group and let H and K be subgroups of G. Prove that H K is a subgroup of G.
Proof: Let H and K be subg
F13MTH 3175 Group Theory (Prof.Todorov)
Quiz 5 (PracticeSol)
Name:
1. Let F : Z Z be a function dened as F (x) = 10x.
(a) Prove that F is a group homomorphism.
Proof:
F is a function F : Z Z (already stated in the problem, so no need to prove it).
F (a
S11MTH 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 G. Prove that the set
aHa1 = cfw_aha1 | where h H is a subgroup of G.
Proof:
Claim 1: aHa1 G, i.e. is a subset of G.
Proof of Claim 1
F13MTH 3175 Group Theory (Prof.Todorov)
Quiz 5 (Practice)
Name:
(Make sure that you explain all of your answers! )
1. Let F : Z Z be a function dened as F (x) = 10x.
(a) Prove that F is a group homomorphism.
(b) Find Ker(F )
(c) Find Im(F )
2. Let f : Z6
S11MTH 3175 Group Theory (Prof.Todorov)
Quiz 4 Practice Solutions
Name:
Dihedral group D4
1. Let D4 =< , t | 4 = e, t2 = e, tt = 1 > be the dihedral group.
(a) Write the Cayley table for D4 . You may use the fact that cfw_e, , 2 , 3 , t, t, t2 , t3
are a
S10MTH 3175 Group Theory (Prof.Todorov)
Quiz 6 (Practice)
Name:
Some of the problems are very easy, some are harder.
1. Let F : Z Z be a function dened as F (x) = 10x.
(a) Prove that F is a group homomorphism.
(b) Find Ker(F )
(c) Find Im(F )
2. Let f : Z
S10MTH 3175 Group Theory (Prof.Todorov) Quiz 5 (Practice-Some Solutions) Name:
1. Let Sn be the group of permutations on n elements cfw_1, 2, 3, . . . , n. Let An be the subgroup
of even permutations. Prove that An is normal subgroup of Sn .
2. Let S15 be
S10MTH 3175 Group Theory (Prof.Todorov)
Quiz 6 (Practice)
Name:
Some of the problems are very easy, some are harder.
1. Let F : Z Z be a function dened as F (x) = 10x.
(a) Prove that F is a group homomorphism.
(b) Find Ker(F )
Solution: Ker(F ) = cfw_0.
P
F13MTH 3175 Group Theory (Prof.Todorov)
Quiz 3 Practice
Name:
1. Let G be a group and let H be a subgroup of G. Let a G. Prove that the set
aHa1 = cfw_aha1 | where h H is a subgroup of G.
2. Consider the permutation =
12345678
S8 .
23718645
(a) Write as
F11MTH3175 GroupTheory
Quiz 1
Please explain all your work.
1. Use Euclidian algorithm:
4pts
(a) Find the greatest common divisor gcd(660, 105) = d.
4pts
(b) Find a pair of integers s, t Z such that s 660 + t 105 = d.
2. Let p, q, r be three distinct prim
F11MTH 3175 Group Theory (Prof.Todorov) Final (Practice)
1
3 PERMUTATIONS
Some Denitions
For your convenience, we recall some of the denitions:
A group G is called simple if it has no proper normal subgroups, i.e. the only normal subgroups
are cfw_e and
S11MTH 3175 Group Theory (Prof.Todorov) Final (Practice Some Solutions) PROPERTIES
2 BASIC
1
Some Denitions
For your convenience, we recall some of the denitions:
A group G is called simple if it has no proper normal subgroups, i.e. the only normal subgr
F11MTH 3175 Group Theory (Prof.Todorov)
Quiz 3 Practice
Name:
Please explain all your work ! When using theorems, write their statements.
1. (a) Find the conjugate of (1234)(56) by a = (25) in S7 .
(b) Find the conjugate of (1234)(56) by a = (27) in S7 .
S11MTH 3175 Group Theory (Prof.Todorov)
Quiz 4 Practice
Name:
1. Consider external direct product: Z6 Z15 .
(a) What is the order of (2, 3)?
(b) What is the order of (2, 12)?
(c) What is the order of (4, 12)?
(d) What is all possible orders of elements (a
MTH 3175 Group Theory (Prof.Todorov)
Quiz 2 (Practice)
Name:
Please explain all your work ! When using theorems, write their statements.
1. Let G be a group and let H and K be subgroups of G. Prove that H K is a subgroup of G.
2. Let G = U (20). Is G cycl
F11MTH 3175 Group Theory (Prof.Todorov)
Quiz 3, Solutions
Name:
1. (a) Find the conjugate of (1234)(56) by a = (25) in S7 .
Denition: A conjugate of by a is a ( ) = aa1 .
Remark: If order of an element a in a group is |a| = n, then a1 = an1 .
If a = (25)
MTH 3175 Group Theory (Prof.Todorov)
Quiz 2 (Practice)
Name:
Please explain all your work ! When using theorems, write their statements.
1. Let G be a group and let H and K be subgroups of G. Prove that H K is a subgroup of G.
Proof: Let H and K be subgro
S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice (Some Solutions) Name:
1. Consider external direct product: Z6 Z15 .
(a) What is the order of (2, 3) Z6 Z15 ?
Answer:
Order of (a, b) is |(a, b)| = lcm(|a|, |b|).
|(2, 3)| = lcm(|2|, |3|) = lcm(3,
F11MTH 3175 Group Theory (Prof.Todorov)
Quiz 6 (Practice)
Name:
Some of the problems are very easy, some are harder.
1. Let G and H be two groups and G H the external direct product of G and H .
(a) Prove that the map f : G H H G dened as f (g, h) = (h, g
S11MTH 3175 Group Theory (Prof.Todorov)
Quiz 5 Practice
Name:
Some problems are really easy, some are harder, some are repetitions.
1. Let Sn be the group of permutations on n elements cfw_1, 2, 3, . . . , n. Let An be the subgroup
of even permutations. P
F11MTH 3175 Group Theory (Prof.Todorov)
Quiz 5 (Practice Solutions)
Name:
Some problems are really easy, some are harder, some are repetitions.
1. Let Sn be the group of permutations on n elements cfw_1, 2, 3, . . . , n. Let An be the subgroup
of even per
F11MTH 3175 Group Theory (Prof.Todorov)
Quiz 6 (PracticeSolutions)
Name:
Some of the problems are very easy, some are harder.
1. Let G and H be two groups and G H the external direct product of G and H .
(a) Prove that the map f : G H H G dened as f (g, h
S11MTH 3175 Group Theory (Prof.Todorov) Final (Practice)
1
3 PERMUTATIONS
Some Denitions
For your convenience, we recall some of the denitions:
A group G is called simple if it has no proper normal subgroups, i.e. the only normal subgroups
are cfw_e and