Algebra II - Groups

Question 1

Define a group

Answer 1

Question 2

When is a group abelian

Answer 2

when it’s commutative ie when ab=ba for all a,b

Question 3

If g,h are elements of a multiplicative group G, what is (gh)^-1

Answer 3

h^-1g^-1

Question 4

Define K^star for a field K

Answer 4

K^star = K \ {0}

Question 5

What is GL_n(K) and SL_n(K)? Are they abelian?

Answer 5

GL_n(K) is the set of invertible n x n matrices with entries in K. SL_n(K) is the set of n x n matrices with determinant 1. They are both non-abelian

Question 6

What is Sym(X)

Answer 6

the set of permutations of X; bijections from X to X

Question 7

Define order of a group G

Answer 7

The number of elements in G, denoted | G |

Question 8

Define the order of an element g in G

Answer 8

The smallest natural number n such that gⁿ = 1. Denoted o(g) or | g |

Question 9

Define an isomorphism between two groups

Answer 9

Question 10

Proposition: Any two infinite cyclic groups are isomorphic. For a positive integer n, any two cyclic groups of order n are isomorphic

Answer 10

Question 11

When is a subset H of a group G a subgroup

Answer 11

When it forms a group under the same operation as G

Question 12

If H is a non-empty subset of a group G, when is it a subgroup (2 conditions)

Answer 12

h,g in H implies gh in H

h in H implies h^-1 in H

Question 13

Lemma: Let G be a group with H,K subgroups of G. Then HnK is also a subgroup of G

Answer 13

Question 14

Define the direct product of two multiplicative groups G,H

Answer 14

G x H is the set { (g,h) | g in G, h in H}

Question 15

What is the Klein Four Group

Answer 15

{1,a,b,c}

a,b,c have order 2, and the product of any two of these elements is the other one

Question 16

What could a group of order 4 be isomorphic to?

Answer 16

Either a cyclic group or the Klein Four group

Question 17

Define the Dihedral group of order 2n

Answer 17

It is the isometries of P, which are

i) n rotations (2(pi)k)/n around the centre of P, denoted a^k
ii) n reflections about lines that pass through the centre of P, denoted b

Then G = { a^k | 0 < k <= n} union {a^kb | 0 < k <= n}

Question 18

In D_2n what is ba?

Answer 18

a^-1b and a^-1 = a^n-1

Question 19

When do elements {g₁,…….,g_r} generate a group G

Answer 19

if every element of G can be obtained by repeated multiplication of the g_i and their inverses

Question 20

Proposition: Let G be a group of order 2n generated by two elements a,b with aⁿ = 1, b² = 1 and ba = a^-1b. Then G is isomorphic to D_2n

Answer 20

Question 21

Proposition: Let G be a group of order mn generated by two elements a and b that satisfy the equations a^m = 1, bⁿ = 1 and ab = ba. Then G is isomorphic to C_m x C_n

Answer 21

Question 22

Define the Quaternion group, Q₈

Answer 22

Question 23

Define the right coset Hg for g in G

Answer 23

{hg | h in H}

Question 24

For g,k in G, give the two equivalent conditions to k in Hg

Answer 24

Hg = Hk

kg^-1 in H

Question 25

State Lagrange’s theorem

Answer 25

Let G be a finite group and H a subgroup of G. Then the order of H divided the order of G. Furthermore |G| = |H| x [G:H]

Question 26

Define the index of H in G

Answer 26

it’s the number of distinct cosets of H in G

Question 27

Proposition: Let G be a finite group. Then for any g in G the order |g| of g divides the order |G| of G

Answer 27

Question 28

Proposition: Let G be a group having prime order p. Then G is cyclic and G is isomorphic to C_p

Answer 28

Question 29

When is a subgroup H of G normal?

Answer 29

If the left and right cosets gH and Hg are equal for all g in G

Question 30

Proposition: If G is any group and H is a subgroup with [G : H] = 2, then H is a normal subgroup of G

Answer 30

Question 31

Proposition: Let H be a subgroup of the group G. Then H is normal in G if and only if ghg^-1 is in H for all g in G and h in H

Answer 31

Question 32

Lemma: Let G be a group in which g² = 1 for all g in G, and let a,b be distinct non-identity elements of G. Then {1, a, b, ab} is a subgroup of order 4

Answer 32

Question 33

Proposition: Let G be a group of order 6. Then G is isomorphic to either C₆ or D₆

Answer 33

Question 34

If N is a normal subgroup of G and Ng, Nh are cosets of N in G, what is (Ng)(Nh)

Answer 34

Ngh

Question 35

Theorem: Let N be a normal subgroup of a group G. Then the set G/N of right cosets Ng of N in G forms a group under multiplication

Answer 35

Question 36

Define a homomorphism between groups

Answer 36

Question 37

Proposition: Let phi: G to H be a homomorphism. Then phi is injective if and only if ker(phi) = {1_G}

Answer 37

Question 38

Proposition: Let phi: G to H be a homomorphism. Then im(phi) is a subgroup of H

Answer 38

Question 39

State the first isomorphism theorem for groups

Answer 39

Question 40

State the second isomorphism theorem for groups

Answer 40

Question 41

Let G be a group and X a set. Define an action of G on X

Answer 41

Question 42

Define the kernel of an action of G on X, when is the action faithful?

Answer 42

Question 43

State Cayley’s theorem

Answer 43

Every group G is isomorphic to a permutation group (a subgroup of Sym(X) for some set X)

Question 44

For an action of G on X, define the orbit

Answer 44

Question 45

What is the stabilisier of x in G?

Answer 45

Stab_G(x) = { g in G | g . x = x}

Question 46

Prove that Stab_G(x) is a subgroup of G, where G acts on X

Answer 46

Question 47

State and prove the Orbit-Stabiliser theorem

Answer 47

Question 48

Define the conjugation action, and both the conjugate classes and centraliser

Answer 48

Question 49

Given a permutation in cyclic notation, how do we obtain the conjugate fgf^-1 of g

Answer 49

Replace each element x in X in the cycles of g by f(x)

Question 50

Define cycle type of a permutation

Answer 50

It has cycle type 2^r₂3^r₃…… if it has exactly r_i cycles of length i

Question 51

Proposition: A₄ has no subgroup of order 6

Answer 51

Question 52

Define a Sylow p-subgroup of G

Answer 52

Question 53

State Sylow’s theorem

Answer 53

Question 54

If G is a group of order pⁿm, define Syl_p(G)

Answer 54

Syl_p(G) = {H is a subgroup of G | |H| = pⁿ}

Question 55

Give two conditions on | Syl_p(G) |

Answer 55

|Syl_p(G)| divides [G:P]

|Syl_p(G)| = 1 if and only if P is a normal subgroup of G

Question 56

Proposition: A simple abelian group is cyclic of prime order

Answer 56

Question 57

Lemma: There are no simple groups of order 24

Answer 57