Group Theory

Question 1

Define a generated group

Answer 1

Let G be a group S be a subset of G. The subgroup of G generated by S is the intersection of all subgroups of G that contain S.

Question 2

Define a finitely generated group

Answer 2

A group G is finitely generated if there exists a finite set S such that <S> = G.</S>

Question 3

Define a word

Answer 3

Given a set S and the inverse elements S^(-1) where S∩S^(-1) = ∅, a word is a finite sequence x1,… , xm where each xi ∈ S∪S^(−1).

Question 4

Define an elementary contraction

Answer 4

A word w’ is an elementary contraction of a word w if all instances of xx^(−1) or x^(-1)x in w are removed.

Question 5

Define a reduced word

Answer 5

A word with no elementary contractions.

Question 6

Define concatenation of words

Answer 6

The concatenation of two words x1x2…xm and y1y2…yn is the word x1x2…xmy1y2…yn.

Question 7

Define a free group

Answer 7

The free group on the set S are the reduced words in the alphabet S under concatenation.

Question 8

Give the universal property of free groups

Answer 8

Given any set S, any group G and any function f : S → G, there is a unique homomorphism ϕ: F(S) → G such that the following diagram commutes: G ←f S i→ F(S) ϕ→ G , where i: S → F(S) is the canonical inclusion.

Question 9

Define normal closure

Answer 9

Let B be a subset of a group G. The normal closure of B is the smallest normal subgroup of G that contains B.

Question 10

Define a presentation

Answer 10

Let X be a set, and let R be a collection of elements of F(X). The group with presentation ⟨X | R⟩ is defined to be F(X)/⟨⟨R⟩⟩.

Question 11

Define the canonical presentation

Answer 11

The canonical presentation for G is ⟨G | R(G)⟩, where R(G) is the kernel of the homomorphism mapping F(G) to the corresponding element in G.

Question 12

State the First Isomorphism Theorem

Answer 12

Given a homomorphism ϕ: G1 → G2, there is a well-defined isomorphism between G1/kerϕ and Imϕ, given by gker(ϕ)→ ϕ(g) and the map ϕ factors into the canonical quotient map G1 → G1/ ker ϕ, the above isomorphism, and the inclusion Imϕ → G2.

Question 13

State the second isomorphism theorem

Answer 13

Let G be a group, and H, N subgroups of G with N ◁ G. Then H∩N ◁ H and HN/N is isomorphic to H/H∩N.

Question 14

State the third isomorphism theorem

Answer 14

Let N, K be normal subgroups of a group G, with K ⊆ N. Then N/K ◁ G/K and there is an isomorphism between (G/K)/(N/K) and G/N.

Question 15

State the classification theorem of finite simple groups

Answer 15

Let G be a finite simple group. Then G is isomorphic to one of the following:
i) A cyclic group of prime order Cp.
ii) A group An for n > 4.
iii) A finite group of Lie type.
iv) An explicit list of 26 exceptions known as the sporadic groups.

Question 16

Define a composition series

Answer 16

A composition series for a group G is a sequence of subgroups {e} = G0 ◁ G1 ◁ · · · ◁ Gn−1 ◁ Gn = G such that each composition factor Gi+1/Gi simple.

Question 17

State Jordan Holder’s Theorem

Answer 17

Let G be a finite group. Then all composition series of G have the same length and the same composition factors, including multiplicities up to permutation.

Question 18

Define a solvable group

Answer 18

A finite group G is solvable if every composition factor of G is a cyclic group of prime order.

Question 19

Define a commutator

Answer 19

[a, b] = a^(−1)b^(−1)ab.

Question 20

Define the commutator subgroup

Answer 20

Let G be a group. The derived subgroup or commutator subgroup of G is the subgroup G’ = ⟨a^(−1)b^(−1)ab such that a, b ∈ G⟩.

Question 21

Define a derived series

Answer 21

Let G be a finite group. Set G(0) = G and for k ≥ 1, define G(k) = (G(k−1))’. The series G = G(0) ≥ G(1) (= G′(2) ≥ … is called the derived series of G.

Question 22

Define a derived length

Answer 22

Given a derived series G(n), if k ≥ 0 is the first instance that G(k) = {e}, then k is known as the derived length of G.

Question 23

Define a semi direct product

Answer 23

Let G be a group, H ≤ G and N ◁ G. Then G is an internal semi-direct product of H and N, denoted G = N ⋊ H, if G = NH and H ∩ N = {e}.

Question 24

Define an extension

Answer 24

Let A and B be groups. Then an extension of A by B is a group G together with a normal subgroup K such that K is isomorphic to A and G/K is isomorphic to B.

Question 25

Define a short exact sequence

Answer 25

Let M, N, P be groups. A short exact sequence involving M, N and P is a sequence {e} → M i→ N π→ P → {e} with i injective, π surjective and Im(i) = ker(π).

Question 26

Define a split

Answer 26

An extension of a group A by a group B described by the short exact sequence {e} → A i→ G π→ B → {e} is said to split if there exists a group homomorphism j : B → G such that π◦j = id.

Question 27

State Cauchy's Theorem

Answer 27

Given a group G and p a prime, if p divides |G|, then G has an element of order p.

Question 28

Define a p-group

Answer 28

Let G be a finite group. We say that G is a p-group, where p is a prime, if the order of G is a power of p.

Question 29

Define a Sylow p-subgroup

Answer 29

Let G a group of order p^a.m where p is a prime that does not divide m. Then we call a subgroup H of G a Sylow p-subgroup if H has order p^a.

Question 30

State Sylow's 1st Theorem

Answer 30

Let G be a finite group, |G| = p^a.m, where p is a prime that does not divide m and a ≥ 0. Then, there is a Sylow p-subgroup of G.

Question 31

State Sylow's 2nd Theorem

Answer 31

Let G be a finite group, |G| = p^a.m, where p is a prime that does not divide m and a ≥ 0. Then, two Sylow p-subgroups of G are conjugate.

Question 32

State Sylow's 3rd Theorem

Answer 32

Let G be a finite group, |G| = p^a.m, where p is a prime that does not divide m and a ≥ 0. Then, the number of Sylow p-subgroups, n_p satisfies n_p ≡ 1 (mod p) and n_p divides m.