# Linear representation theory of generalized dihedral groups

This article gives specific information, namely, linear representation theory, about a family of groups, namely: generalized dihedral group.
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This article discusses the linear representations of the generalized dihedral group corresponding to a finite abelian group $H$. This group is defined as:

$G := \langle H,x \mid x^2 = e, xhx = h{^-1} \ \forall \ h \in H \rangle$.

## The irreducible representations

For the discussion below, we let $n$ denote the order of $H$. We let $S$ denote the set of squares in $H$, and $K = H/S$ is the quotient group. $K$ is an elementary abelian $2$-group, and we denote its order by $2^k$.

### One-dimensional representations

There are $2^{k+1}$ of these, described as follows.

Since $K$ is an elementary abelian group of order $2^k$, it has $2^k$ one-dimensional representations. Each of these gives rise to a one-dimensional representation of $H$, by composing with the quotient map from $H$ to $K$. Further, each such representation takes values $\pm 1$.

For every such representation $\rho$ of $H$, there are two corresponding one-dimensional representations of $G$ whose restriction to $H$ is $\rho$: one representation sends the element $x$ to $(1)$, and the other representation sends $x$ to $-1$. Thus, we get a total of $2^{k+1}$ one-dimensional representations.

### Two-dimensional irreducible representations

These two-dimensional representations arise from all the representations of $H$ that do not descend to $K$. There are $n - 2^k$ of these.

Start with any representation $\rho$ of $H$ that does not have $S$ in its kernel. Consider the induced representation to $G$. Then, this induced representation is irreducible. Further, the induced representations for two representations are equal if and only if they are complex conjugates of each other -- this can readily be verified by looking at character values.

Since $\rho$ does not descend to $K$, it is not equal to its complex conjugate. Thus, we obtain $(n - 2^k)/2$ inequivalent two-dimensional irreducible representations this way.