Definition:Klein Four-Group
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Group Example
The Klein $4$-group, often denoted $K_4$, is a group of $4$ elements, each of which is self-inverse.
Cayley Table
The Klein $4$-group can be described completely by showing its Cayley table:
$\quad \begin {array} {c|cccc} & e & a & b & c \\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & e & a \\ c & c & b & a & e \\ \end {array}$
Subgroups
Also known as
The Klein four-group (or Klein's four-group) is also known as the four-group or the Viergruppe.
Hence it is often denoted $V$.
The term is not always hyphenated::
Some sources refer to it as the dihedral group of order $4$.
Also see
- Results about the Klein four-group can be found here.
Source of Name
This entry was named for Felix Christian Klein.
Internationalization
Four-group is translated:
| In German: | Viergruppe | (literally: four-group) |
Sources
- 1965: J.A. Green: Sets and Groups ... (previous): Tables: $3$. Alternating group $\map A 4$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.5$: Example $15$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26 \iota$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups: Exercise $6$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 44$ Some consequences of Lagrange's Theorem
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Klein four-group
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Groups with four elements
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Klein's four group
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Klein four-group