𝑝-groups have unbounded realization multiplicity

Berg, Jen; Schultz, Andrew

doi:10.1090/S0002-9939-2014-11967-8

$p$-groups have unbounded realization multiplicity
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by Jen Berg and Andrew Schultz

Proc. Amer. Math. Soc. 142 (2014), 2281-2290

DOI: https://doi.org/10.1090/S0002-9939-2014-11967-8

Published electronically: March 11, 2014

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Abstract:

In this paper we interpret the solutions to a particular Galois embedding problem over an extension $K/F$ satisfying $\operatorname {Gal}(K/F) \simeq \mathbb {Z}/p^n\mathbb {Z}$ in terms of certain Galois submodules within the parameterizing space of elementary $p$-abelian extensions of $K$; here $p$ is a prime. Combined with some basic facts about the module structure of this parameterizing space, this allows us to exhibit a class of $p$-groups whose realization multiplicity is unbounded.

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