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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This particular group is the smallest (in terms of order): non-cyclic group
The Klein-four group is defined in the following equivalent ways:
- It is the direct product of the group with itself
- It is the group comprising the elements under coordinate-wise multiplication
- It is the unique non-cyclic group of order 4
- It is the subgroup of the symmetric group on 4 elements comprising the double transpositions, and the identity element.
This particular group is Abelian
The automorphism group is naturally identified with the group as follows. Each element of the automorphism group corresponds to a permutation of the three non-identity elements.
The non-automorphism endomorphisms include:
- The trivial map
- Pick an arbitrary direct sum decomposition and an arbitrary two-element subgroup. Then the projection on the first direct factor for the decomposition, composed with the isomorphism to the other two-element subgroup, is an endomorphism.
All subgroups are normal, since the group is Abelian. There is a total of five subgroups: the whole group, the trivial subgroup, and two-element subgroups (viz copies of the cyclic group of order 2).
The Klein-four group is a characteristically simple group, since it is a direct power of a simple group. Hence, the only characteristic subgroups are the trivial subgroup and the whole group.