Module VII·Article I·~1 min read
Normal Subgroups and Factor Groups
Group Theory: Sylow Theorems
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Normal Subgroups
A subgroup N is called normal (N ⊲ G) if gN = Ng for all g ∈ G, that is, gNg⁻¹ = N.
Equivalently: N is invariant with respect to all inner automorphisms.
Examples: {e} and G are always normal. In an abelian group — any subgroup is normal. The center Z(G) = {g: gx = xg ∀x} is normal.
The kernel of a homomorphism ker φ is always a normal subgroup.
Factor Group
If N ⊲ G, the cosets gN form a group G/N with the operation (gN)(hN) = (gh)N.
|G/N| = |G|/|N| (index of the subgroup).
Homomorphism Theorem: If φ: G → H is a homomorphism, then G/ker φ ≅ Im φ.
Simple Groups
A group is simple if the only normal subgroups are {e} and G itself. Simple groups are the “atoms” of group theory.
Finite abelian simple groups: ℤ/pℤ (p is prime).
The alternating group Aₙ (even permutations) is simple for n ≥ 5. This is a key fact in the proof of the unsolvability of the general equation of degree ≥ 5 in radicals (Abel–Ruffini theorem).
Classification of Finite Simple Groups
This is a grand theorem of the 20th century: every finite simple group is either ℤ/pℤ, or the alternating Aₙ (n ≥ 5), or a group of Lie type (series of classical groups over finite fields), or one of the 26 sporadic (exceptional) groups. The proof took thousands of pages by hundreds of mathematicians (1955–2004).
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