Let $\varphi : G \to \text{GL}(m, \mathbb{C})$ be an irreducible representation. Its degree, or equivalently the degree of its (linear) character, is by definition $m$.

I was wondering whether the following is true: if $m$ is any (non-zero) integer, does there exist a finite group $G$ and an irreducible representations $\varphi: G \to \text{GL}(m, \mathbb{C})$? In other words, can every integer occur as the degree of an irreducible representation?

If the answer is yes, is there also a set of groups and representations $\{G_i, \varphi_i\}_{i\in\mathbb{N}}$ known for which $\varphi_i$ has degree $i$?

References to source material are very welcome.


2 Answers 2


Yes, for instance the symmetric group $S_{n+1}$ has an irreducible representation of degree $n$. Precisely, take the canonical permutation representation $V$ of $S_{n+1}$; it has degree $n+1$, and we can write $V=V'\oplus L$ where $L$ is a copy of the trivial representation and $V'$ is irreducible, and of degree $n$. It is called the standard representation of $S_{n+1}$.


Let $G_p$ represent an extraspecial $p$-group of order $p^3,$ $p$ prime. There are two non-isomorphic types of such groups, but that does not matter: the character degrees (over $\mathbb{C}$) are the same: $G_p$ has $p^2$ linear characters and $p-1$ irreducible characters of degree $p$. That last fact is of use, for if $m=p_1^{r_1} \cdot p_2^{r_2} \cdots p_k^{r_k}$ is the decomposition in primes, and for a group $G$, $G^{(i)}$ denotes the direct product of $i$ copies of $G$, and $A$ is an arbitrary finite abelian group, then the group $$G_{p_1}^{(r_1)} \times G_{p_2}^{(r_2)} \cdots \times G_{p_k}^{(r_k)} \times A$$ has an irreducible character of degree $m$. In fact, it has $\Pi_{i=1}^k (p_i-1)^{r_i}$ irreducible characters of degree $m$. The order of this group is $m^3 \cdot |A|$.

Note If $m$, and for linear characters $1$, are the only occurring irreducible characters degrees of $G$, then it can be shown that at least one of the following occurs:

(a) $G$ has an abelian normal subgroup of index $m$
(b) $m=p^e$, $p$ prime and $G$ is the direct product of a $p$-group and an abelian group.

For details see I.M. Isaacs, Character Theory of Finite Groups, Theorem (12.5) ff.


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