Faithful group action

From Groupprops
Revision as of 02:22, 9 December 2008 by Vipul (talk | contribs) (New page: {{group action property}} ==Definition== ===Definition in action terms=== A group action of a group <math>G</math> on a set <math>S</math> is termed '''faithful''' or '''effective''' if...)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article defines a group action property or a property of group actions: a property that can be evaluated for a group acting on a set.
View a complete list of group action properties|Get help on group action property lookup|Get exploration suggestions
VIEW RELATED: group action property implications | group action property non-implications | {{{context space}}} metaproperty satisfactions | group action metaproperty dissatisfactions | group action property satisfactions |group action property dissatisfactions

Definition

Definition in action terms

A group action of a group G on a set S is termed faithful or effective if for any non-identity elemnet g \in G, there is s \in S such that g.s \ne s.

Definition in terms of homomorphisms

A group action of a group G on a set S is termed faithful or effective if the corresponding homomorphism from G to \operatorname{Sym}(S) is an injective homomorphism.