Template:Dihedral group of twice prime order

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The group is defined as the dihedral group of degree {{{2}}} and hence order {{{1}}}. It is given explicitly by the presentation:

{{{3}}}

Here, $e$ denotes the identity element.

Group properties

General properties

Property	Satisfied	Explanation
abelian group	No	In the given presentation, $x$ and $a$ do not commute.
nilpotent group	No	Dihedral groups are nilpotent if and only if their order is a power of two.
solvable group	Yes	Dihedral groups are solvable
simple group	No	There are proper non-trivial normal subgroups, for example the subgroup generated by $x$ in the given presentation.

Basic arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	{{{1}}}	groups with same order"{{{" can not be assigned to a declared number type with value 1.
exponent of a group	{{{1}}}	groups with same order and exponent of a group"{{{" can not be assigned to a declared number type with value 1. | groups with same exponent of a group"{{{" can not be assigned to a declared number type with value 1.
derived length	2	groups with same order and derived length"{{{" can not be assigned to a declared number type with value 1. | groups with same derived length	Derived series goes through Klein four-group of double transpositions.
nilpotency class	--	--	not a nilpotent group.
minimum size of generating set	2	groups with same order and minimum size of generating set"{{{" can not be assigned to a declared number type with value 1. | groups with same minimum size of generating set	Not cyclic so greater than 1. {{{3}}} is a generating set. Hence is 2.

GAP implementation

Group ID

This finite group has [[groups of order {{{1}}}|order {{{1}}}]] and has ID 1 among the groups of order {{{1}}} in GAP's SmallGroup library. For context, there are groups of order {{{1}}}. It can thus be defined using GAP's SmallGroup function as:

SmallGroup({{{1}}},1)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup({{{1}}},1);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [{{{1}}},1]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

Description	Functions used
DihedralGroup({{{1}}})	DihedralGroup