Quotient Group

Question 1

What is a quotient set?

Answer 1

Let G be a group, and H be a normal subgroup of G. Then the set of all left cosets of H in G is the quotient set. That is:

G/H = { aH : a ∈ G}

Question 2

What is a quotient group?

Answer 2

Let G be a group, H be a normal subgroup of G, and G/H be a quotient set. Then G/H can be made a group by the operation ⊙ defined as:
aH ⊙ bH = (ab)H for all a, b ∈ G. This group is known as the quotient group.

Question 3

To prove (G/H, ⊙) is a group, show that: it is closed under ⊙.

Answer 3

Since G is a group, and a, b ∈ G, then ab ∈ G.
Let ā = aH, ƀ = bH
Recall G/H = {aH : a ∈ G}
Then ā⊙ƀ = aH ⊙ bH = (ab)H
This, G/H is closed.

Question 4

What is a commutator?

Answer 4

Let a, b be elements of a group G, then a commutator is the element aba^-1b^-1.

Question 5

What is a derived group G’?

Answer 5

If G is a group, then G’ is the subgroup of all the commutators of G.
That is, G’ = {aba^-1b^-1: a, b € G}

Question 6

What is a normaliser?

Answer 6

Let G be a group and H a subgroup of G. The normaliser of H then is a set of all the elements a € G such that aH = Ha.
N_G(H) = {a € G: aH = Ha}

Question 7

Is the normaliser of a subgroup empty?

Answer 7

No, because e € G, and eH = He. So e € N_G(H)

Question 8

Define a normal subgroup with respect to normalizers.

Answer 8

H is normal = {a € G: aH = Ha , for all a € G}
This implies that H is a subgroup for which every element in G is its normalizer. If H is normal in G, N_G(H) = G

Question 9

What is the proof of the first isomorphism theorem?

Answer 9

Here