## 18 Jun A Short Note on Certain Vector Spaces Associated with Finite Groups

Throughout this article $G$ is a finite group.

1. Free vector space generated by $G$ : Given a field $\mathbb{F}$ we take the set of all formal linear combinations of form $\displaystyle{\sum_{g\in G}}a_g g$ where $a_g\in \mathbb{F}$ $\forall g\in G$. We define addition on the set as
$\displaystyle{\sum_{g\in G}}a_g g+\displaystyle{\sum_{g\in G}}b_g g=\displaystyle{\sum_{g\in G}}(a_g+b_g) g$
and scalar multiplication as
$\alpha \times (\displaystyle{\sum_{g\in G}}a_g g)=\displaystyle{\sum_{g\in G}}(\alpha \times a_g) g$
for all $\alpha \in\mathbb{F}$. We take $\displaystyle{\sum_{g\in G}}0_{\mathbb{F}} g$ to be the zero in the set. This set now forms a vector space called the free vector space generated by $G$ over $\mathbb{F}$. Clearly, the dimension of this vector space is equal to the order of the $G$.
2. Permutation Representation : Let $X$ be a set with a finite number of elements. Let $G$ act on $X$. To this set $X$ we can associate a vector space spanned by a basis, where each basis element corresponds to a unique element in $X$. Hence we can index the basis elements with the elements of $X$, i.e. let the basis be $(e_x)_{x\in X}$. We can now define a $G$-action on this basis and extend this action by $G$-linearity to the entire vector space. The action is
$g.e_x=e_{g.x}$
We call this vector space a permutation representation of $G$.
3. Class Function : A function $f:G\rightarrow \mathbb{F}$, where $\mathbb{F}$ is a field said to be a class function if
$f(hgh^{-1})=f(g)$
for all $g,h in G$.

The set of all class functions for the given group forms a vector space $\mathscr{V}$ over $\mathbb{F}$. What is the dimension of this vector space? We observe that every class function is invariant on the conjugacy classes of $G$. Hence an easy guess would be that the dimension of this vector space is equal to the number of conjugacy classes of $G$. After one makes this guess it is very easy to see why it is so by producing a basis for $\mathscr{V}$.

Certain Remarks

• One can notice that we haven’t used the group structure in 1. Hence, we can extend this definition to any set with a finite cardinality. Infact we can further extend this definition to any set Y where we define formal linear combinations to be $\displaystyle{\sum_{y\in Y}}a_y y$ where atmost finitely many of the $a_y$‘s can be non-zero.
• In 1. we can generalize the construction to make a module over a ring in a similar way. We can infact make a ring by defining the multiplication in a way similar to the multiplication of polynomials. We call the group so formed as the group ring or the group algebra.
• In 3. if the underlying field is algebraically closed and its characteristic doesn’t divide the order of $G$(this is not the strongest condition though) then the characters of all irreducible representations of $G$(all of which are class functions) form an orthogonal basis for the vector space with respect to the following inner product :
$=\displaystyle{\sum_{g\in G}}f_1(g)f_2(g^{-1})$
where $f_1$ and $f_2$ are class functions.

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