# Groups with no small generating set

Is there a classification of groups having the property that any set of $d$ elements (say including the identity) is contained in a proper subgroup?

It is appealing to call the maximum such integer (when finite) some sort of "dimension" or measure of being "not cyclic". As one example, we have elementary abelian groups of order $p^d$. I haven't thought much about nonabelian groups, but there are examples.

Disclaimers: Apologies if this is either a trivial classification or not of much research interest. It is perhaps well-studied already, but I don't know what to call this property. I thought I'd try to ask, since this is a group-theoretic analog of something else which interests me. Should I try group pub forum instead?

• My guess is that this is a very hard question. One relevant fact is that every finite simple group can be generated by $2$ elements; see mathoverflow.net/questions/59213/…. – Andy Putman Jan 8 '14 at 18:16
• There are general algebraic considerations with algebras having minimal generating sets of (at least) $d+1$ elements. Other than free spectra in locally finite varieties, I do not know of any work using it though. (Many varieties of groups are not locally finite, and I do not know the group theory literature well enough to suggest additional terminology or useful references.) Gerhard "Check The Universal Algebra Literature" Paseman, 2014.01.08 – Gerhard Paseman Jan 8 '14 at 18:21
• If a quotient of $G$ has large rank, then $G$ itself has large rank, so you can take any extension of a large-rank group and get another one. This is not particularly interesting, so you might want to restrict attention to groups for which any proper quotient has smaller rank. ("Just-rank-k, groups"?) Some obvious examples of this type are simple groups and powers of $\mathbb{Z}_p$. Note that for finite simple $G$, $G^2$ also has rank $2$, so it's not a group of this type. For more examples, you want taking the abelianization to decrease rank, so it's natural to look at perfect groups. – Anton Malyshev Jan 9 '14 at 2:01