Groups of order 20

This article gives information about, and links to more details on, groups of order 20
See pages on algebraic structures of order 20 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 20. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 20
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 20
subgroup structure	subgroup structure of groups of order 20
linear representation theory	linear representation theory of groups of order 20 projective representation theory of groups of order 20 modular representation theory of groups of order 20
endomorphism structure, automorphism structure	endomorphism structure of groups of order 20
group cohomology	group cohomology of groups of order 20

Statistics at a glance

The number 20 has prime factors 2 and 5. The prime factorization is as follows:

$\!20=2^{2}\cdot 5^{1}=4\cdot 5$

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^{a}q^{b}$ -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Quantity	Value	Explanation
Total number of groups up to isomorphism	5	See classification of groups of order four times a prime congruent to 1 modulo four.
Number of abelian groups up to isomorphism	2	(number of abelian groups of order $2^{2}$ ) $\times$ (number of abelian groups of order $5^{1}$ ) = (number of unordered integer partitions of 2) $\times$ (number of unordered integer partitions of 1) = $2\times 1=2$ . See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism	2	(number of groups of order 4) $\times$ (number of groups of order 5) = $2\times 1=2$ . See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group. See also nilpotent of cube-free order implies abelian.
Number of solvable groups up to isomorphism	5	There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^{a}q^{b}$ -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.
Number of simple groups up to isomorphism	0	All groups of this order are solvable

The list

Group	Second part of GAP ID (GAP ID is (20,second part))	abelian?
dicyclic group:Dic20	1	No
cyclic group:Z20	2	Yes
general affine group:GA(1,5)	3	No
dihedral group:D20	4	No
direct product of Z10 and Z2	5	Yes

Ways to distinguish the groups up to isomorphism

Orders of elements

Further information: element structure of groups of order 20

The groups of order 20 can be differentiated via having different element orders:

Group	orders of elements
dicyclic group:Dic20	1, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 10, 10, 10, 10
cyclic group:Z20	1, 2, 4, 4, 5, 5, 5, 5, 10, 10, 10, 10, 20, 20, 20, 20, 20, 20, 20, 20
general affine group:GA(1,5)	1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5
dihedral group:D20	1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 5, 5, 10, 10, 10, 10
direct product of Z10 and Z2	1, 2, 2, 2, 5, 5, 5, 5, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Number of subgroups

Further information: subgroup structure of groups of order 20

The groups of order 20 are not all able to be distinguished by their number of subgroups. dicyclic group:Dic20 and direct product of Z10 and Z2 both have the same number of subgroups, 10. These two groups can be distinguished via other means, such as abelianness.

Group	number of subgroups
dicyclic group:Dic20	10
cyclic group:Z20	6
general affine group:GA(1,5)	14
dihedral group:D20	22
direct product of Z10 and Z2	10

Number of normal subgroups

The groups, however, all have distinct numbers of normal subgroups:

Group	number of normal subgroups
dicyclic group:Dic20	5
cyclic group:Z20	6
general affine group:GA(1,5)	4
dihedral group:D20	7
direct product of Z10 and Z2	10

Subgroups

Sylow subgroups

Group	Sylow 2-subgroup isomorphism class	Sylow 5-subgroup isomorphism class
dicyclic group:Dic20	cyclic group:Z4	cyclic group:Z5
cyclic group:Z20	cyclic group:Z4	cyclic group:Z5
general affine group:GA(1,5)	cyclic group:Z4	cyclic group:Z5
dihedral group:D20	Klein four-group	cyclic group:Z5
direct product of Z10 and Z2	Klein four-group	cyclic group:Z5