On the representation dimension of finite $p$-groups

Abstract: For a finite group $G$, the representation dimension $δ(G)$ is the least dimension of a faithful complex representation of $G$. We prove that $δ(H\times K) = δ(H) + δ(K)$ whenever $H$ and $K$ are $p$-groups for a fixed prime $p$, and show by example that this additivity can fail more generally for nilpotent groups. We also determine $δ(G)$ for several important classes of non-abelian $p$-groups, {\it viz.} VZ groups, Camina groups, and metacyclic groups; and compute $δ(G)$ for all groups of order $p<sup>6$ ($p\geq 5$), organized by their isoclinism family.

• The paper proves that representation dimension is exactly additive for direct products of finite p-groups over the same prime, while mixed-prime nilpotent products can fall below the expected sum.
• It derives explicit formulas for VZ, Camina, and metacyclic p-groups using faithful irreducible-character sets, central-rank arguments, and maximal character degrees.
• It determines the representation dimension for every isoclinism family of groups of order p^6 for p ≥ 5, revealing a possible dependence on isoclinism and center structure that remains unproved in general.

The representation dimension $\delta(G)$ of a finite group $G$ is the least dimension of a faithful complex representation of $G$, equivalently the smallest $n$ such that $G$ embeds into $GL_n(\mathbb{C})$. This paper by Kaur, Kulshrestha, and Udeep (2608.18814) makes three contributions: it establishes additivity of $\delta$ for direct products of $p$-groups over a fixed prime, shows that this additivity fails for general nilpotent groups, and computes $\delta(G)$ explicitly for VZ groups, Camina groups, metacyclic groups, and all groups of order $p^6$ ($G$0), organized by isoclinism family.

Background and method

The paper works entirely at the level of characters. A faithful character of least degree decomposes as a sum of distinct irreducible characters whose kernels intersect trivially, with no proper subcollection having trivial kernel intersection; such a set is said to afford $G$1. Two structural facts drive most arguments. First, a result of Bardestani–Mallahi-Karai–Salmasian states that for a non-abelian $G$2-group, any set affording $G$3 has cardinality equal to $G$4, the rank of the center. Second, Mann's proposition on normally monomial groups (groups whose irreducible characters are induced from linear characters of normal subgroups) implies that if a faithful irreducible character is induced from a normal subgroup $G$5, then $G$6 is abelian of maximal order among abelian subgroups, and its degree equals $G$7. The trivial upper bound $G$8 follows by summing the pullback characters $G$9.

Additivity for direct products

The first main result shows that for normally monomial $G$0-groups $G$1 and $G$2 with cyclic centers, any faithful character of least degree of $G$3 decomposes as $G$4 with $G$5 and $G$6. The proof proceeds by an exhaustive case analysis on the possible degrees of two irreducible constituents: cases where either constituent exceeds the corresponding maximum violate the upper bound arithmetically, while cases where a constituent falls below the maximum force all central elements of order $G$7 into its kernel — via the observation that a low-degree induced character must come from a non-abelian normal subgroup whose derived subgroup meets the center — so the two kernels cannot have trivial intersection.

This is then strengthened to a clean structural theorem: for any two $G$8-groups $G$9 and $n$0 over a fixed prime $n$1, $n$2. The proof adapts Wright's technique: writing $n$3, $n$4, a minimal faithful character of $n$5 has exactly $n$6 irreducible constituents, and one shows that after relabeling, the first $n$7 constituents restrict to a faithful character of $n$8 and the last $n$9 to one of $G$0. The key combinatorial step identifies each constituent's kernel intersection with $G$1 as a hyperplane spanned by all but one of $G$2 independent order-$G$3 central elements. Consequently, additivity extends to arbitrary finite direct products of $G$4-groups over a fixed prime.

Additivity fails across distinct primes. The paper gives explicit counterexamples for nilpotent groups. For instance, taking $G$5 of order 16 with $G$6 and $G$7 a non-abelian group of order 27, one has $G$8 but $G$9 (verified computationally). Further examples show failure even when $GL_n(\mathbb{C})$0 is abelian but not cyclic: an extraspecial group of order 8 times $GL_n(\mathbb{C})$1 gives $GL_n(\mathbb{C})$2, and $GL_n(\mathbb{C})$3 gives $GL_n(\mathbb{C})$4. These examples delineate precisely when the upper bound is not tight.

On the positive side, additivity survives when $GL_n(\mathbb{C})$5 is cyclic (of arbitrary order): if $GL_n(\mathbb{C})$6 is a non-abelian $GL_n(\mathbb{C})$7-group with cyclic center and $GL_n(\mathbb{C})$8 or $GL_n(\mathbb{C})$9 ($\delta$0), or more generally a normally monomial $\delta$1-group with cyclic center, then $\delta$2. In both proofs, the argument forces the minimal faithful character to consist of exactly one linear character and one character of maximal degree, since characters of smaller degree kill a fixed central element of order $\delta$3 while linear characters contain the derived subgroup in their kernels.

Closed-form formulas for structured classes

For VZ $\delta$4-groups (all nonlinear irreducibles vanish off the center), where $\delta$5, the paper proves

$\delta$6

using a lower bound from kernel-rank considerations and matching it with an existence result for quasi-permutation representations. For Camina $\delta$7-groups, which have nilpotency class at most 3, the formula simplifies to

$\delta$8

since every constituent of a minimal faithful character lies above the center and has degree $\delta$9. For metacyclic $p$0-groups, the answer depends on cyclicity of the center: if $p$1 is cyclic, $p$2 (a faithful irreducible exists); if $p$3 is non-cyclic, $p$4, realized by one linear plus one faithful nonlinear constituent.

Groups of order $p$5

For $p$6, there are $p$7 groups of order $p$8, distributed over 43 isoclinism families. The paper determines $p$9 for every family:

Isoclinic families	$\delta(G)$0
$\delta(G)$1	$\delta(G)$2, $\delta(G)$3, $\delta(G)$4, or $\delta(G)$5
$\delta(G)$6	$\delta(G)$7, $\delta(G)$8, or $\delta(G)$9
$p^6$0, $p^6$1	$p^6$2 or $p^6$3
$p^6$4, $p^6$5	$p^6$6 or $p^6$7
$p^6$8	$p^6$9 or $G$00
$G$01	$G$02
$G$03, $G$04	$G$05
$G$06, $G$07	$G$08
$G$09, $G$10	$G$11
$G$12, $G$13	$G$14
$G$15	$G$16

Families $G$17 through $G$18 are handled uniformly: they satisfy $G$19 except $G$20 with $G$21, giving $G$22 or $G$23 respectively. The harder families ($G$24–$G$25, $G$26–$G$27, $G$28) require case-by-case analysis combining lower bounds from the Bardestani et al. lemma with explicit witness sets of irreducible characters drawn from the presentations of O'Brien–Prajapati–Udeep. Groups that decompose as a nontrivial direct product are handled via the additivity theorem together with known values for orders up to $G$29; groups of order $G$30 and $G$31 can be computed directly with the GAP function EmbeddingDegree.

A notable empirical pattern emerges from this table: within these families, isoclinic groups with isomorphic centers have equal representation dimension. The authors state this observation but concede they cannot prove it in general.

Limitations and open questions

Several boundaries of the results deserve note. The additivity theorem applies only to $G$32-groups over a fixed prime; the counterexamples show it genuinely fails for nilpotent groups mixing primes, and the paper does not attempt a classification of when equality holds in the mixed-prime case beyond the cyclic-factor results. The formulas for VZ and Camina groups rely on prior existence results for faithful sets of irreducible characters, and the metacyclic computation assumes the standard parametrized presentation. Two problems are left open: computing $G$33 for special $G$34-groups generally (extending the extraspecial case), and determining whether isoclinic groups of equal order with isomorphic centers always share the same representation dimension — a question suggested but not resolved by the order-$G$35 data.

Conclusion

The paper settles the behavior of representation dimension under direct products for $G$36-groups, proving exact additivity and delimiting its failure for nilpotent groups through concrete examples, and delivers complete closed-form computations for VZ, Camina, and metacyclic $G$37-groups as well as an exhaustive determination of $G$38 across all isoclinism families of groups of order $G$39 for $G$40. The observed dependence of $G$41 only on the isoclinism class and center structure stands as the principal unproven regularity emerging from the computations.