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$G$-Set (Definition)

If $ G$ is a group and $ X$ a set, then $ X$ is called a left $ G$-Set if there exists a mapping $ \lambda : G \times X \rightarrow X$ with

$\displaystyle \lambda(g_1,\lambda(g_2,x)) = \lambda(g_1 g_2, x) $

or shorter with $ \lambda(g,x) = gx$

$\displaystyle g_1(g_2(x)) = (g_1 g_2)(x) $

for all $ x \in X$ and $ g_1,g_2 \in G$. And when $ G$ acts on a set $ X$, the set $ X$ is always a $ G$-set.

$ X$ is called a right $ G$-Set if there exists a mapping $ \lambda : X \times G \rightarrow X$ with

$\displaystyle \lambda(\lambda(x,g_2),g_1) = \lambda(x,g_2 g_1) $



"$G$-Set" is owned by jwaixs.
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See Also: group action
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Cross-references: right, acts on, mapping, group

This is version 2 of $G$-Set, born on 2008-03-20, modified 2008-04-05.
Object id is 10423, canonical name is GSet.
Accessed 146 times total.

Classification:
AMS MSC: 20-00 (Group theory and generalizations :: General reference works )
Pending Errata and Addenda
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