𝓜

Math Notes

University Mathematics

ConceptComplete

Simple Groups and Classification

Simple groups are the "atoms" of group theory. The classification of finite simple groups is one of the monumental achievements of 20th-century mathematics.

Simple Groups

Definition10.4Simple group

A group GG is simple if G≠{e}G \neq \{e\} and the only normal subgroups of GG are {e}\{e\} and GG itself.

ExampleFamilies of simple groups
  1. Cyclic: Z/pZ\mathbb{Z}/p\mathbb{Z} for prime pp (the only abelian simple groups).
  2. Alternating: AnA_n for n≥5n \geq 5 (∣A5∣=60|A_5| = 60, the smallest non-abelian simple group).
  3. Lie type: PSLn(q)\mathrm{PSL}_n(q), PSp2n(q)\mathrm{PSp}_{2n}(q), PSOn(q)\mathrm{PSO}_n(q), etc. For example, PSL2(7)≅GL3(F2)\mathrm{PSL}_2(7) \cong \mathrm{GL}_3(\mathbb{F}_2) has order 168.
  4. Sporadic: 26 exceptional simple groups, including the Mathieu groups M11,M12,M22,M23,M24M_{11}, M_{12}, M_{22}, M_{23}, M_{24} and the Monster group M\mathbb{M} of order approximately 8×10538 \times 10^{53}.

The Classification Theorem

Theorem10.2Classification of finite simple groups

Every finite simple group is isomorphic to one of:

  1. A cyclic group Z/pZ\mathbb{Z}/p\mathbb{Z} of prime order.
  2. An alternating group AnA_n (n≥5n \geq 5).
  3. A group of Lie type (16 infinite families).
  4. One of the 26 sporadic groups.

The proof spans tens of thousands of pages across hundreds of journal articles, contributed by over 100 mathematicians from the 1950s to 2004.

Simplicity of AnA_n

Theorem10.3$A_n$ is simple for $n \\geq 5$

The alternating group AnA_n is simple for all n≥5n \geq 5.

Proof

The key step is showing that every normal subgroup N⊴AnN \trianglelefteq A_n with N≠{e}N \neq \{e\} contains all 3-cycles. Since 3-cycles generate AnA_n, this forces N=AnN = A_n.

If σ∈N∖{e}\sigma \in N \setminus \{e\}, consider the commutator [σ,τ]=στσ−1τ−1[\sigma, \tau] = \sigma\tau\sigma^{-1}\tau^{-1} for various 3-cycles τ\tau. For n≥5n \geq 5, one can always find τ\tau such that [σ,τ][\sigma, \tau] is a nontrivial element of NN with smaller support. Iterating, NN contains a 3-cycle. Since all 3-cycles are conjugate in AnA_n (for n≥5n \geq 5), NN contains all 3-cycles. ■\blacksquare

■
RemarkThe Monster and Moonshine

The Monster group M\mathbb{M} is the largest sporadic simple group with ∣M∣=246⋅320⋅59⋅76⋅112⋅133⋅17⋅19⋅23⋅29⋅31⋅41⋅47⋅59⋅71|\mathbb{M}| = 2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71. The unexpected connections between the Monster and modular functions (Monstrous Moonshine, proved by Borcherds in 1992) revealed deep ties between group theory, number theory, and mathematical physics.