Quotient Group Flashcards
What is a quotient set?
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}
What is a quotient group?
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.
To prove (G/H, ⊙) is a group, show that: it is closed under ⊙.
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.
What is a commutator?
Let a, b be elements of a group G, then a commutator is the element aba-1b-1.
What is a derived group G’?
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}
What is a normaliser?
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.
NG(H) = {a € G: aH = Ha}
Is the normaliser of a subgroup empty?
No, because e € G, and eH = He. So e € NG(H)
Define a normal subgroup with respect to normalizers.
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, NG(H) = G