Variety of groups is congruence-uniform: Difference between revisions
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{{variety of groups satisfaction | {{variety of groups property satisfaction}} | ||
==Statement== | ==Statement== | ||
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In the language of groups, the above statement interprets as: all the cosets of a [[normal subgroup]] have the same size. | In the language of groups, the above statement interprets as: all the cosets of a [[normal subgroup]] have the same size. | ||
==Related facts== | |||
===For analogous algebraic structures=== | |||
Some similar algebraic structures for which the variety is congruence-uniform: | |||
* [[Variety of loops is congruence-uniform]] | |||
* [[Variety of Lie rings is congruence-uniform]] | |||
Some similar algebraic structures for which the variety is not congruence-uniform: | |||
* [[Variety of monoids is not congruence-uniform]] | |||
==Facts used== | |||
# [[uses::Left cosets are in bijection via left multiplication]] | |||
# [[uses::First isomorphism theorem]] | |||
==Proof== | ==Proof== | ||
This follows from a more general fact for a group: the [[left cosets are in bijection via left multiplication]]. | ===Proof idea=== | ||
This follows from a more general fact for a group: the [[left cosets are in bijection via left multiplication]], combined with the fact that for any congruence class on a group, the congruence classes are the cosets of a normal subgroup. This is essentially the statement of the [[first isomorphism theorem]]. | |||
Latest revision as of 11:59, 21 August 2011
This article gives the statement, and possibly proof, of a property satisfied by the variety of groups
View a complete list of such property satisfactions
Statement
The variety of groups is a congruence-uniform variety. In other words, every group is a congruence-uniform algebra in the variety of groups. More explicitly, given any group and any congruence on it, all the congruence classes are of equal size.
Translation to the language of groups
In the language of groups, the above statement interprets as: all the cosets of a normal subgroup have the same size.
Related facts
For analogous algebraic structures
Some similar algebraic structures for which the variety is congruence-uniform:
Some similar algebraic structures for which the variety is not congruence-uniform:
Facts used
Proof
Proof idea
This follows from a more general fact for a group: the left cosets are in bijection via left multiplication, combined with the fact that for any congruence class on a group, the congruence classes are the cosets of a normal subgroup. This is essentially the statement of the first isomorphism theorem.