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The permutation representation on the left cosets $G / H$

Example

Let $k$ be a field, let $G$ be a finite group, and let $H \leq G$ . Left multiplication on the coset set $G / H$ gives a permutation representation of $G$ on the vector space $k^{(G / H)}$ .

Facts & Assumptions

Given: A field $k$ , a finite group $G$ , and a subgroup $H \leq G$ .

[L1]

The left cosets are the subsets $x H = {x h : h \in H}$ of $G$ (Left and right cosets $g H$ and $H g$ of a subgroup).

[L2]

A finite $G$ -set gives a permutation representation on the free vector space with that basis (The trivial representation, the regular representation, and permutation representations from finite $G$ -sets).

Verification

technique · direct

1.1L1given

For $g, x \in G$ , left multiplication sends the coset $x H$ to $(g x) H$ , so $G$ acts on the finite set $G / H$ by $g \cdot (x H) : = (g x) H$ .

2.1step 1.1L2∎

By [L2], the induced representation on the basis vectors $e_{x H}$ of $k^{(G / H)}$ is $g \cdot e_{x H} = e_{(g x) H}$ . So the matrix of $g$ in this basis is a permutation matrix recording the induced permutation of the cosets.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Webb, A Course in Finite Group Representation Theory, Example 4.3.4 (standard reference, not scraped)

The permutation representation on the left cosets G/HG/H