p-GROUPS WITH CYCLIC OR GENERALISED QUATERNION HUGHES SUBGROUPS: CLASSIFYING TIDY p-GROUPS

Let G be a p-group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the...

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Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society 2023-12, Vol.108 (3), p.443-448
Main Authors: BEIKE, NICOLAS F., CARLETON, RACHEL, COSTANZO, DAVID G., HEATH, COLIN, LEWIS, MARK L., LU, KAIWEN, PEARCE, JAMIE D.
Format: Article
Language:English
Subjects:
Classification
Hypotheses
Quaternions
Subgroups
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Summary:Let G be a p-group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the Hughes subgroup of G is generalised quaternion, then G must be generalised quaternion. With these results in hand, we classify the tidy p-groups.
ISSN:0004-9727
1755-1633
DOI:10.1017/S000497272300031X