Alternating group:A4

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Definition

Permutation definition

The alternating group A_4 is defined as the group of all even permutations on a set of 4 elements.

Presentation

Group properties

Solvability

This particular group is solvable

The commutator subgroup of A_4 is the Klein group of order 4, namely the normal subgroup comprising double transpositions. This is Abelian.

Thus, A_4 is solvable of solvable length 2, or in other words, it is a metabelian group.

Nilpotence

This particular group is not nilpotent

Abelianness

This particular group is not Abelian

Simplicity

This particular group is not simple

Since A_4 has a proper nontrivial commutator subgroup, it is not simple.