# elementary abelian group

An *elementary abelian group* is an abelian group^{} in which every non-trivial element has the same finite order. It is easy to see that the non-trivial elements must in fact be of prime order, so every elementary abelian group is a $p$-group (http://planetmath.org/PGroup4) for some prime $p$.

Elementary abelian $2$-groups are sometimes called *Boolean groups*.
A group in which every non-trivial element has order $2$ is necessarily Boolean, because abelianness is automatic: $xy={(xy)}^{-1}={y}^{-1}{x}^{-1}=yx$.
There is no analogous result for odd primes, because for every odd prime $p$ there is a non-abelian group^{} of order ${p}^{3}$ and exponent $p$.

Let $p$ be a prime number.
Any elementary abelian $p$-group can be considered as a vector space over the field of order $p$, and is therefore isomorphic^{} to the direct sum^{} of $\kappa $ copies of the cyclic group^{} of order $p$, for some cardinal number^{} $\kappa $. Conversely, any such direct sum is obviously an elementary abelian $p$-group.
So, in particular, for every infinite^{} cardinal $\kappa $ there is, up to isomorphism^{}, exactly one elementary abelian $p$-group of order $\kappa $.

Title | elementary abelian group |
---|---|

Canonical name | ElementaryAbelianGroup |

Date of creation | 2013-03-22 14:53:11 |

Last modified on | 2013-03-22 14:53:11 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 12 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20F50 |

Classification | msc 20K10 |

Defines | elementary abelian |

Defines | Boolean group |