# Problems in Group Theory

See the attached file.

Suppose that G is a finite group and H is a normal subgroup of G.

(i) Explain briefly how the quotient group G/H is defined. [You should explain what the elements of G/H are, how the multiplication is defined, what the identity element is and how inverses are defined. You do not need to prove that the multiplication is well-defined.]

Show that the map

?:G?G/H

?(g)=gH

is a homomorphism. What is the kernel of ??

(ii) Hence show that if ?:G/H?GL(n,C) is a representation of the quotient group G/H then the map ? ?:G?GL(n,C) defined by

? ?(g)=?(gH)

(for each g?G) is a representation of G.

(iii) Now let G=A_4, the alternating group of even permutations of {1,2,3,4}. Let

H={e,(12)(34),(13)(24),(14)(23)}

Show that H?G. To which well-known group is the quotient group G/H isomorphic?

(iv) For G and H as in (iii) above, write down (without proof) three degree one representations of G/H.

(v) Hence describe three degree 1 (and hence irreducible) representations of G=A_4. Write down the characters of the representations you have found on representatives of the four conjugacy classes of G.

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#### Solution Summary

The solution solves several problems in group theory involving such topics as group representations, alternating groups, permutation groups, group homomorphisms, conjugacy classes, normal subgroups, general linear groups, and characters.

Group Theory

1.

i. State the axioms for an equivalence relation

ii. The relation n mod 3 divides the non-negative integers

(i.e, n in Z such that n ≥ 0) into how many partitions?

Show that n = 0 mod 3 is an equivalence relation.

2. Prove that, for any matrices, A, B and C:

A+B=B+A

And:

A+(B+C)=(A+B)+C

( i.e., that the matrix addition is both commutative and associative)

For simplicity, prove these properties using 2x2 matrices.

3. Prove that addition modulo n, written + is:

i. Associative.

ii. Commutative.

There are two ways to prove these properties. Each way requires a definition or two:

i. For n ≥ 2, 0 ≤ a, b ≤ n+1,

a+ b= a+b if a+b< n

a+n-n if a+b≥ n

ii. Writing a for a mod n and (a+b) = a+ b, then:

(p+ q) ≡ (p +q )

Do the proof using both methods. Which is more "algebraic" (in the sense of "abstract" algebra)?

4. Prove that addition modulo n, written + is:

i. Associative

ii Commutative.

( extra definations required : a for a mod n and (pà?q) = pà? q, so

(pà? q) (p à?q )

5.

i. State the axioms defining a group

- If (Z, +) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative.

- If (Z, à?) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z, identify the inverse. Also show that + is associative.

- If (R, à?) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative.

iii. In each case, deterimine whether the algebraic structure is a group. For each such group:

- show how it satifies the group axioms

- Draw the cayley table for the group and list the inverse elements

i. For S=(0,2), a+2b ≡ (a+b) mod 2 and aà?2b ≡ (a à?b)mod2

a. (S,+ ) ( possibly an additive group)

b. (S,∙ 2) (possibly a multiplicative group).

ii. For S = (0,1,2) where n=2,3 and + and à? are defined as in the last part.

a. (S,+n) ( possibly an additive group)

b. (S,∙ n) (possibly a multiplicative group).

Determine whether any of the groups is an abelian group. If any of them are abelian:

i. state the conditions under which a group is abelian

ii. show that the group is abelian

6. there are only two groups of order four (Z 4and v). How many groups are there of the order five? Draw cayley tables for each one of them( the element should be named a,b,c,d,e) is either (or both) of the groups of order four subgroup of any of those of order five? If so which one

7. for each of the following structures, state whethere it is a group. If it is, state whether it is abelian or not.

i.For any set, A, the set of one-to one and onto functions, f: A →A under composition ( written "◦").

ii.The set of all subsets of the three-element set (a,b,c) ( there are eight such subsets) under:

a. Union

b. Intersection

iii. The set G=(a+b√5| a,b in Q) under addition and multiplication

iv The set consisting of non-zero numbers under

a. addition

b. division

v. The set (1,5,7,11) under multiplication modulo 12. Draw cayley table

vi. The set (4,6) under multiplication modulo 12. draw cayley table

vii. The set of real numbers under à?, where aà?b = 2(a+b)

viii The set of real numbers under +, where a+b = a+b-10

ix. The set of rotational symmetries of a regular hexagon under composition

x The following sets of permutations under composition

i. (e,(12),(123),(1234))

ii. (e,(12), (34), (12), (34))

8.Let G be a group, (G, * ) in which there is an element, a , such that g * g=g . prove that g=e

9.Prove that for every element, a, of a group, G, the order of a and a^-1 are the same ( including the case of an infinite order)

10.Let x and y be elements of a group, G. Prove that the elements xy and yx have the same orders

11.Find the subgroups of

i. Z7

ii. Z8

iii Z9

12.

i. determine which of the folowign are subgroups of under +

a. (0)

b. (-1,0,1)

c. (n| n=10m for some integer m

d.(p| p is a prime number

e. (0,1,2,3,4) under addition modulo 5

ii. Determine which of the following are subgroups of under mulitiplication:

a. (1, -1)

b. (x |x=3, for some integer n

c. (x |x=p/2ⁿ for some integers, p,n)

d. (x| x=k 3 for some interger k

Group Theory

Group Theory

1.

i. State the axioms for an equivalence relation

ii. The relation n mod 3 divides the non-negative integers

(i.e, n in Z such that n ≥ 0) into how many partitions?

Show that n = 0 mod 3 is an equivalence relation.

2. Prove that, for any matrices, A, B and C:

A+B=B+A

And:

A+(B+C)=(A+B)+C

( i.e., that the matrix addition is both commutative and associative)

For simplicity, prove these properties using 2x2 matrices.

3. Prove that addition modulo n, written + is:

i. Associative.

ii. Commutative.

There are two ways to prove these properties. Each way requires a definition or two:

i. For n ≥ 2, 0 ≤ a, b ≤ n+1,

a+ b= a+b if a+b< n

a+n-n if a+b≥ n

ii. Writing a for a mod n and (a+b) = a+ b, then:

(p+ q) ≡ (p +q )

Do the proof using both methods. Which is more "algebraic" (in the sense of "abstract" algebra)?

4. Prove that addition modulo n, written + is:

i. Associative

ii Commutative.

( extra definations required : a for a mod n and (pà?q) = pà? q, so

(pà? q) (p à?q )

5.

i. State the axioms defining a group

- If (Z, +) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative.

- If (Z, à?) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z, identify the inverse. Also show that + is associative.

- If (R, à?) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative.

iii. In each case, deterimine whether the algebraic structure is a group. For each such group:

- show how it satifies the group axioms

- Draw the cayley table for the group and list the inverse elements

i. For S=(0,2), a+2b ≡ (a+b) mod 2 and aà?2b ≡ (a à?b)mod2

a. (S,+ ) ( possibly an additive group)

b. (S,∙ 2) (possibly a multiplicative group).

ii. For S = (0,1,2) where n=2,3 and + and à? are defined as in the last part.

a. (S,+n) ( possibly an additive group)

b. (S,∙ n) (possibly a multiplicative group).

Determine whether any of the groups is an abelian group. If any of them are abelian:

i. state the conditions under which a group is abelian

ii. show that the group is abelian

6. there are only two groups of order four (Z 4and v). How many groups are there of the order five? Draw cayley tables for each one of them( the element should be named a,b,c,d,e) is either (or both) of the groups of order four subgroup of any of those of order five? If so which one

7. for each of the following structures, state whethere it is a group. If it is, state whether it is abelian or not.

i.For any set, A, the set of one-to one and onto functions, f: A →A under composition ( written "◦").

ii.The set of all subsets of the three-element set (a,b,c) ( there are eight such subsets) under:

a. Union

b. Intersection

iii. The set G=(a+b√5| a,b in Q) under addition and multiplication

iv The set consisting of non-zero numbers under

a. addition

b. division

v. The set (1,5,7,11) under multiplication modulo 12. Draw cayley table

vi. The set (4,6) under multiplication modulo 12. draw cayley table

vii. The set of real numbers under à?, where aà?b = 2(a+b)

viii The set of real numbers under +, where a+b = a+b-10

ix. The set of rotational symmetries of a regular hexagon under composition

x The following sets of permutations under composition

i. (e,(12),(123),(1234))

ii. (e,(12), (34), (12), (34))

8.Let G be a group, (G, * ) in which there is an element, a , such that g * g=g . prove that g=e

9.Prove that for every element, a, of a group, G, the order of a and a^-1 are the same ( including the case of an infinite order)

10.Let x and y be elements of a group, G. Prove that the elements xy and yx have the same orders

11.Find the subgroups of

i. Z7

ii. Z8

iii Z9

12.

i. determine which of the folowign are subgroups of under +

a. (0)

b. (-1,0,1)

c. (n| n=10m for some integer m

d.(p| p is a prime number

e. (0,1,2,3,4) under addition modulo 5

ii. Determine which of the following are subgroups of under mulitiplication:

a. (1, -1)

b. (x |x=3, for some integer n

c. (x |x=p/2ⁿ for some integers, p,n)

d. (x| x=k 3 for some interger k