Everything about Klein Four-group totally explained
In
mathematics, the
Klein four-group (or just
Klein group or
Vierergruppe, often symbolized by the letter
V) is the
group Z
2 × Z
2, the
direct product of two copies of the
cyclic group of
order 2 (or any
isomorphic variant). It was named
Vierergruppe by
Felix Klein in his
Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade in 1884.
The Klein four-group is the smallest non-cyclic group. The only other group with four elements, up to isomorphism, is the cyclic group of order four: Z
4 (see also the
list of small groups).
All elements of the Klein group (except the
identity) have order 2.
It is
abelian, and isomorphic to the
dihedral group of order 4.
The Klein group's
Cayley table is given by:
» : with the action being multiplication modulo 8.
Field
The Klein four-group is isomorphic to the additive group of
finite field GF(4):
+ | 0 1 A B · | 0 1 A B
--+
--+
0 | 0 1 A B 0 | 0 0 0 0
1 | 1 0 B A 1 | 0 1 A B
A | A B 0 1 A | 0 A B 1
B | B A 1 0 B | 0 B 1 A
In Popular Culture
The Klein Four is an
a cappella singing group at
Northwestern University, best known for their song "Finite SImple Group (of Order Two)."
Further Information
Get more info on 'Klein Four-group'.
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