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$G$-module (Definition)

Let $V$ a vector space over some field $K$ (usually $K=\mathbbmss{Q}$ or $K=\mathbbmss{C}$). Let $G$ be a group which acts on $V$. This means that there is an operation $\psi \colon G\times V \to V$ such that

  1. $gv \in V$.
  2. $g(hv) = (gh)v$
  3. $ev = v$
where $gv$ stands for $\psi(g,v)$ and $e$ is the identity element of $G$.

If in addition,

\begin{displaymath}g(cv + dw) = c(gv)+d(gw)\end{displaymath}

for any $g\in G$, $v,w \in V$, $c,d\in K$, we say that $V$ is a $G$-module. This is equivalent with the existence of a group representation from $G$ to $GL(V)$.



"$G$-module" is owned by rspuzio. [ full author list (2) | owner history (3) ]
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See Also: group representation, group

Keywords:  group, representation
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Cross-references: group representation, equivalent, addition, identity element, operation, acts on, group, field, vector space
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This is version 3 of $G$-module, born on 2005-01-26, modified 2006-10-18.
Object id is 6663, canonical name is GModule.
Accessed 1482 times total.

Classification:
AMS MSC: 20C99 (Group theory and generalizations :: Representation theory of groups :: Miscellaneous)
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