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什么是 simple group？

Definition 1.31 University Mathematics — Year 3 · 第 1 章 — Group Theory

A group $G \neq \{e\}$ is simple if its only normal subgroups are $\{e\}$ and $G$ . A nonabelian simple group is not solvable: $D(G) \trianglelefteq G$ is not $\{e\}$ (else $G$ abelian), so $D(G) = G$ and the derived series is constant. The abelian simple groups are exactly the $\Z/p\Z$ , $p$ prime (an abelian group is simple iff it has no proper nontrivial subgroup, iff it is cyclic of prime order by Lagrange).

The ten subgroups of the dihedral group D_4 = r, s r4 = s2 = e,\ srs-1 = r-1. The three subgroups of index 2 (middle row) are normal, as is the center r2 (highlighted); the four reflection subgroups fall into two conjugacy classes of two. Chains from bottom to top give composition series, e.g. \e\ r2 r D_4: factors ℤ/2ℤ, ℤ/2ℤ, ℤ/2ℤ — always the same multiset, as Jordan–Hölder demands. — The ten subgroups of the dihedral group $D_4 = \langle r, s \mid r^4 = s^2 = e,\ srs^{-1} = r^{-1}\rangle$ . The three subgroups of index $2$ (middle row) are normal, as is the center $\langle r^2\rangle$ (highlighted); the four reflection subgroups fall into two conjugacy classes of two. Chains from bottom to top give composition series, e.g. $\{e\} \trianglelefteq \langle r^2\rangle \trianglelefteq \langle r\rangle \trianglelefteq D_4$ : factors $\Z/2\Z, \Z/2\Z, \Z/2\Z$ — always the same multiset, as Jordan–Hölder demands.

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