Elementary finite groups and families catalogue

Core idea

50 separately identified objects and parameterized families. Individual finite groups and constrained families retain distinct entries. Orders are exact integers or formulas; simplicity is stated under each entry's parameter restrictions. An isomorphism is a recorded relationship, not a reason to collapse the entries.

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Finite abelian invariant factors

Object	Order	Simplicity	Kind of entry
$C_{d_1}\times\cdots\times C_{d_r}$	$\prod_{i=1}^r d_i$	parameter-dependent	parameterized family

Finite alternating

Object	Order	Simplicity	Kind of entry
$A_{1}$	$1$	not simple	specified object
$A_{2}$	$1$	not simple	specified object
$A_{3}$	$3$	simple under the stated constraints	specified object
$A_{4}$	$12$	not simple	specified object
$A_{5}$	$60$	simple under the stated constraints	specified object
$A_{6}$	$360$	simple under the stated constraints	specified object
$A_{7}$	$2520$	simple under the stated constraints	specified object
$A_{8}$	$20160$	simple under the stated constraints	specified object
$A_{n}$	$n!/2$	simple under the stated constraints	parameterized family

Finite cyclic

Object	Order	Simplicity	Kind of entry
$C_{1}$	$1$	not simple	specified object
$C_{2}$	$2$	simple under the stated constraints	specified object
$C_{3}$	$3$	simple under the stated constraints	specified object
$C_{n}$	$n$	parameter-dependent	parameterized family
$C_{p}$	$p$	simple under the stated constraints	parameterized family

Finite dihedral

Object	Order	Simplicity	Kind of entry
$D_{2}$	$2$	simple under the stated constraints	specified object
$D_{2n}$	$2n$	parameter-dependent	parameterized family
$D_{4}$	$4$	not simple	specified object
$D_{6}$	$6$	not simple	specified object

Finite elementary abelian

Object	Order	Simplicity	Kind of entry
$C_p^{1}$	$p$	simple under the stated constraints	parameterized family
$C_p^{2}$	$p^{2}$	not simple	parameterized family
$C_p^{3}$	$p^{3}$	not simple	parameterized family
$C_p^{n}$	$p^{n}$	parameter-dependent	parameterized family

Finite generalized quaternion

Object	Order	Simplicity	Kind of entry
$Q_{16}$	$16$	not simple	specified object
$Q_{2^n}$	$2^n$	not simple	parameterized family
$Q_{32}$	$32$	not simple	specified object
$Q_{8}$	$8$	not simple	specified object

Finite gl

Object	Order	Simplicity	Kind of entry
$\mathrm{GL}_{1}(q)$	$q-1$	parameter-dependent	parameterized family
$\mathrm{GL}_2(2)$	$6$	not simple	specified object
$\mathrm{GL}_2(3)$	$48$	not simple	specified object
$\mathrm{GL}_{2}(q)$	$(q^2-1)(q^2-q)$	not simple	parameterized family
$\mathrm{GL}_{3}(q)$	$(q^3-1)(q^3-q)(q^3-q^2)$	parameter-dependent	parameterized family
$\mathrm{GL}_{n}(q)$	$\prod_{j=0}^{n-1}(q^n-q^j)$	parameter-dependent	parameterized family

Finite heisenberg

Object	Order	Simplicity	Kind of entry
$H_1(\mathbb F_q)$	$q^3$	not simple	parameterized family
$H_2(\mathbb F_q)$	$q^5$	not simple	parameterized family
$H_3(\mathbb F_q)$	$q^7$	not simple	parameterized family
$H_n(\mathbb F_q)$	$q^{2n+1}$	not simple	parameterized family

Finite klein four

Object	Order	Simplicity	Kind of entry
$V_4$	$4$	not simple	specified object

Finite sl

Object	Order	Simplicity	Kind of entry
$\mathrm{SL}_{1}(q)$	$1$	not simple	parameterized family
$\mathrm{SL}_2(2)$	$6$	not simple	specified object
$\mathrm{SL}_2(3)$	$24$	not simple	specified object
$\mathrm{SL}_{2}(q)$	$q(q^2-1)$	parameter-dependent	parameterized family
$\mathrm{SL}_{3}(q)$	$q^3(q^3-1)(q^2-1)$	parameter-dependent	parameterized family
$\mathrm{SL}_{n}(q)$	$\frac{1}{q-1}\prod_{j=0}^{n-1}(q^n-q^j)$	parameter-dependent	parameterized family

Finite symmetric

Object	Order	Simplicity	Kind of entry
$S_{1}$	$1$	not simple	specified object
$S_{2}$	$2$	simple under the stated constraints	specified object
$S_{3}$	$6$	not simple	specified object
$S_{4}$	$24$	not simple	specified object
$S_{6}$	$720$	not simple	specified object
$S_{n}$	$n!$	parameter-dependent	parameterized family

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Elementary finite groups and families catalogue

Core idea

Finite abelian invariant factors

Finite alternating

Finite cyclic

Finite dihedral

Finite elementary abelian

Finite generalized quaternion

Finite gl

Finite heisenberg

Finite klein four

Finite sl

Finite symmetric

Long lists of objects

How to compare objects

Worked comparisons

Low dimensions and the magic square

Category conventions

Data and proof status

Separate algebra-class additions