PlanetMath: faithful group action

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faithful group action

(Definition)

Let be a -set, that is, a set acted upon by a group with action $\psi:G\times A\to A$ . Then for any $g\in G$ , the map $m_g\colon A\to A$ defined by

$\displaystyle m_g(x)= \psi(g,x)$

is a permutation of

(in other words, a bijective function from

to itself) and so an element of

. We can even get an homomorphism from

by the rule $g\mapsto m_g$ .

If for any pair $g,h\in G$ $g\neq h$ we have $m_g\neq m_h$ , in other words, the homomorphism $g\to m_g$ being injective, we say that the action is faithful.

"faithful group action" is owned by rspuzio. [ full author list (4) | owner history (3) ]

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Cross-references: faithful, injective, homomorphism, even, bijective function, words, permutation, map, action, group

This is version 5 of faithful group action, born on 2003-10-15, modified 2005-07-26.
Object id is 5259, canonical name is FaithfulGroupAction.
Accessed 1366 times total.

Classification:

AMS MSC:	20M30 (Group theory and generalizations :: Semigroups :: Representation of semigroups; actions of semigroups on sets)
	16W22 (Associative rings and algebras :: Rings and algebras with additional structure :: Actions of groups and semigroups; invariant theory)

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