# An Invitation to Inverse Group Theory

- Published in 2018
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In group theory there are many constructions which produce a new group from a given one. Often the result is a subgroup: the derived group, centre, socle, Frattini subgroup, Hall subgroup, Fitting subgroup, and so on. Other constructions may produce groups in other ways, for example quotients (solvable residual, derived quotient) or cohomology groups (Schur multiplier). Inverse group theory refers to problems in which a construction and the resulting group is given and we want information about the possible original group or groups; examples are the {\em inverse Schur multiplier problem} (given a finite abelian group is it the Schur multiplier of some finite group?), or the {\em inverse derived group} (given a group $G$ is there a group $H$ such that $H'=G$?). In 1956 B. H. Neumann sent a first invitation to inverse group theory, but apparently the topic did not receive the attention it deserves, so that we attempt here at repeating that invitation. Many of the inverse group problems associated with the constructions referred to above are trivial, but some are not. Like Neumann we will work mainly on inverse derived groups. We also explain how the main questions about inverse Frattini subgroups have been settled. An integral of a group $G$ is a group $H$ such that the derived group of $H$ is $G$. Our first goal is to prove a number of general facts about the integrals of finite groups, and to raise some open questions. Our results concern orders of non-integrable groups (we give a complete description of the set of such numbers), the smallest integral of a group (in particular, we show that if a finite group is integrable it has a finite integral), and groups which can be integrated infinitely often, a problem already tackled by Neumann. We also consider integrals of infinite groups. Regarding inverse Frattini, we explain Neumann's and Eick's results.

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@article{AnInvitationtoInverseGroupTheory, title = {An Invitation to Inverse Group Theory}, abstract = {In group theory there are many constructions which produce a new group from a given one. Often the result is a subgroup: the derived group, centre, socle, Frattini subgroup, Hall subgroup, Fitting subgroup, and so on. Other constructions may produce groups in other ways, for example quotients (solvable residual, derived quotient) or cohomology groups (Schur multiplier). Inverse group theory refers to problems in which a construction and the resulting group is given and we want information about the possible original group or groups; examples are the {\{}\em inverse Schur multiplier problem{\}} (given a finite abelian group is it the Schur multiplier of some finite group?), or the {\{}\em inverse derived group{\}} (given a group {\$}G{\$} is there a group {\$}H{\$} such that {\$}H'=G{\$}?). In 1956 B. H. Neumann sent a first invitation to inverse group theory, but apparently the topic did not receive the attention it deserves, so that we attempt here at repeating that invitation. Many of the inverse group problems associated with the constructions referred to above are trivial, but some are not. Like Neumann we will work mainly on inverse derived groups. We also explain how the main questions about inverse Frattini subgroups have been settled. An integral of a group {\$}G{\$} is a group {\$}H{\$} such that the derived group of {\$}H{\$} is {\$}G{\$}. Our first goal is to prove a number of general facts about the integrals of finite groups, and to raise some open questions. Our results concern orders of non-integrable groups (we give a complete description of the set of such numbers), the smallest integral of a group (in particular, we show that if a finite group is integrable it has a finite integral), and groups which can be integrated infinitely often, a problem already tackled by Neumann. We also consider integrals of infinite groups. Regarding inverse Frattini, we explain Neumann's and Eick's results.}, url = {http://arxiv.org/abs/1803.10179v2 http://arxiv.org/pdf/1803.10179v2}, year = 2018, author = {Jo{\~{a}}o Ara{\'{u}}jo and Peter J. Cameron and Francesco Matucci}, comment = {}, urldate = {2018-04-19}, archivePrefix = {arXiv}, eprint = {1803.10179}, primaryClass = {math.GR}, collections = {unusual-arithmetic} }