Exponent-critical groups

William Cocke, Andrew Misseldine, Geetha Venkataraman and I have a new paper on arXiv, ‘Exponent-critical groups’: arxiv.org/abs/2401.07834, We are interested in the question of when the exponent of a finite group fails to be determined by the exponents of its proper subgroups. A first stab at a definition: the exponent of a group G is larger than the lcm of the exponents of proper subgroups. But this is never true, unless G=C_p. This is boring. So we actually define a group to be exponent-critical if its exponent is larger than the lcm of the exponents of its proper *non-abelian* subgroups. We characterise such groups, extending the classical characterisation of minimal abelian groups due to Miller and Moreno in 1903. The original motivation for the problem came from computational problems in the theory of varieties of groups. I’ve really enjoyed working on this problem. (Geetha came to visit me last summer, and got me interested then.)

Distinct differences in the free group

My PhD student Emma Smith, my former student Dr Luke Stewart, and I have posted a new paper ‘Subsets of free groups with distinct differences’ to arXiv today: https://arxiv.org/abs/2307.03627.

A DDC (distinct difference configuration) is a subset of a group such that the differences $g^{-1}h$ for all distinct $g,h\in D$ are different. The main theorem is:

Let $F_X$ be a free group, freely generated by a set $X$ of cardinality $n$, where $n\geq 2$. Let $m(n,d)$ be the maximum cardinality of a DDC of diameter at most $d$ in $F_X$. As $d\rightarrow\infty$ with $n$ fixed, then
\[
m(n,d)= (2n-1)^{d/3+O(\log d)}
\]
(where the implicit constants might depend on $n$).

Here ‘diameter’ means diameter in the Cayley graph.

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