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Let $G$ be a finite $p$-group and $K$ a field of characteristic $p$. Then the group algebra $KG$ is local and thus the quotient of a non-commutative polynomial ring $K\langle x_i\rangle$ by an admissible ideal: $KG \cong K\langle x_i\rangle/I$.

It is known that the number of minimal generators $x_i$ is given by $ \log_p |G/\Phi(G)|$, see for example the answer of Benjamin Steinberg in Obtaining quiver and relations for finite p-groups.

Question: Is there there a bound or even an explicit formula in terms of group theoretic data for the number of minimal generators of an admissible ideal $I$ such that $KG \cong K\langle x_i\rangle/I$ where we have $ \log_p |G/\Phi(G)|$ variables $x_i$?

Here admissible simply means that $J^n \subseteq I \subseteq J^2$ for some $n \geq 2$ with $J$ the ideal generated by the variables $x_i$.

For example for the quaternion group one can generate $I$ by two elements:

$KG \cong K\langle x_1,x_2\rangle/I$ with $$I=\langle x_1 x_2+x_2 x_1+x_2^2, x_1^2+x_1 x_2+x_2 x_1+x_1^2 x_2+x_1^3 x_2\rangle.$$

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    $\begingroup$ The number of generators of $I$ as a one sided ideal is the dimension of $H^2(G,K)$, so this is an upper bound for the number of generators as a two sided ideal. There are, of course, good bounds on the dimension of $H^2$. $\endgroup$
    – Dave Benson
    Commented Sep 8, 2023 at 15:20
  • $\begingroup$ @DaveBenson Thanks, I guess this holds more generally for any local algebra by the result of Bongartz which gives the number of minimal relations as $dim Ext_A^2(S,S)$. Do you know a group algebra example where we do not have equality? $\endgroup$
    – Mare
    Commented Sep 9, 2023 at 15:17
  • $\begingroup$ I've never really thought about it. It's a good question. $\endgroup$
    – Dave Benson
    Commented Sep 9, 2023 at 18:17

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