Abstract Algebra Midterm 2

Question 1

Cosets

Answer 1

Let H be a subgroup of G and x ∈ G then Hx = {hx | h ∈ H } is the right coset of H in G and xH = { xh | h ∈ H } is the left coset o of H in G.

Question 2

Index

Answer 2

Let G be a group and H be a subgroup of G.
The set of right Cosets of H in G is; G divided by H.
Denoted by | G : H |.
| G : H | is the index of H in G.

• | G : H | is the number of right cosets of H in G

Question 3

Corollary 5.2 - Cosets, partion G

Answer 3

Let G be any group and let H be a subgroup of G. Then the right cosets of H in G partition G.

Question 4

Group Action

Answer 4

Let G be a group and let Ω be a set.
Then G acts on Ω if for all g ∈ G and σ ∈ Ω
there exists g . σ ∈ Ω
such that
1) e . σ = σ for all σ ∈ Ω
2) g . (h . σ ) = (gh) . σ for all g,h ∈ G and for all σ ∈ Ω

Question 5

Transposition

Answer 5

A length 2 cycle (2-cycle) is a transposition.

Question 6

Lemma 5.6- G group, H subgroup of G, TFAE

Answer 6

Let G be a group and let H be a subgroup of G
let x,y ∈ G. Then the following are equivalent:

(a) Hx = Hy (this does NOT mean x=y)
(b) y ∈ Hx
(c) y = hx for some h ∈ H
(d) yx^{-1} ∈ H

in particular Hg = H if and only if g ∈ H

Question 7

The Alternating Group

Answer 7

Let n be a positive integer. A_{n} is the alternating group of degree n so A_{n} is the set of all even permutations in S_{n}

Question 8

Corollary 3.8- A_{n} subgroup of

Answer 8

A_{n} is a subgroup of S_{n}