What does it mean for a subgroup H of G to be normal?
H is normal in G if gH = Hg for all g ∈ G. Left and right cosets are the same, and H is denoted H ◁ G. Normal subgroups allow formation of quotient groups.
What is the kernel of a group homomorphism φ: G → G’?
Ker(φ) = { g ∈ G | φ(g) = e’ }, where e’ is the identity in G’. The kernel measures elements that map to identity and is always a normal subgroup of G.
What is true about the cosets of the kernel of a homomorphism?
For Ker(φ), left and right cosets coincide: gKer(φ) = Ker(φ)g for all g ∈ G. This property guarantees the kernel is normal.
When is left coset multiplication (aH)(bH) = (ab)H well-defined?
Left coset multiplication is well-defined if and only if H is normal in G. Otherwise, the product could depend on the representatives chosen.
What is the factor group (quotient group) of G by H?
G/H is the set of cosets of H in G with multiplication (aH)(bH) = (ab)H. It forms a group when H is normal, allowing study of G “mod H”.
If H is normal in G, how can we define a homomorphism to the quotient group?
The map π: G → G/H defined by π(x) = xH is a homomorphism with kernel H. It sends each element of G to its corresponding coset.
State the Fundamental Homomorphism Theorem.
If φ: G → G’ is a homomorphism with kernel H, then φ[G] is a group, and μ: G/H → φ[G] given by μ(gH) = φ(g) is an isomorphism. Every homomorphism factors through its kernel.
Give four equivalent conditions for H ◁ G.
1) ghg⁻¹ ∈ H for all g ∈ G, h ∈ H. 2) gHg⁻¹ = H for all g ∈ G. 3) There exists a homomorphism φ: G → G’ with Ker(φ) = H. 4) gH = Hg for all g ∈ G.
What is an automorphism of a group G?
An automorphism is an isomorphism from G to itself that preserves the group structure. Automorphisms form a group, Aut(G), under composition.
What is the inner automorphism i_g of G?
i_g(x) = gxg⁻¹ for all x ∈ G. Conjugating x by g is applying i_g. Inner automorphisms are defined using elements of the group itself.
How is normality related to inner automorphisms?
H is normal if and only if i_g[H] = H for all g ∈ G. This means conjugating H by any element of G leaves the set H unchanged, though individual elements may move.
What is a conjugate subgroup?
K is a conjugate of H if K = i_g[H] = gHg⁻¹ for some g ∈ G. Conjugate subgroups are “the same up to an inner automorphism” and share many properties with H.
