# What is meant by a Klein 4 group?

## What is meant by a Klein 4 group?

Klein four group is the symmetry group of a rhombus (or of a rectangle, or of a planar ellipse), with the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180 degree rotation. It is also the automorphism group of the graph with four vertices and two disjoint edges.

## Is the Klein 4 group cyclic?

The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group of order four, up to isomorphism, the cyclic group of order 4. The smallest non-abelian group is the symmetric group of degree 3, which has order 6.

Is the Klein 4 group normal?

The Klein 4-group is an Abelian group. It is the smallest non-cyclic group. It is the underlying group of the four-element field. , and, of course, is normal, since the Klein 4-group is abelian.

Which subgroups are isomorphic to the Klein 4 group?

The Klein four-group is isomorphic to (Z2 × Z2,+) and to (G × G,·). It follows the group (G×G×G,·).

### What is the structure of the Klein four group?

View specific information (such as linear representation theory, subgroup structure) about this group The Klein four-group, usually denoted , is defined in the following equivalent ways: It is the subgroup of the symmetric group of degree four comprising the double transpositions, and the identity element.

### Is the Klein four group the Burnside group?

The Klein four-group, usually denoted , is defined in the following equivalent ways: It is the subgroup of the symmetric group of degree four comprising the double transpositions, and the identity element. It is the Burnside group : the free group on two generators with exponent two.

Which is an endomorphism in Klein four group?

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. trivia group (1 time), cyclic group:Z2 (3 times, all in the same automorphism class), Klein four-group (1 time).