TY - JOUR

T1 - Maximal linear groups induced on the Frattini quotient of a p-group

AU - Bamberg, John

AU - Glasby, S. P.

AU - Morgan, Luke

AU - Niemeyer, Alice C.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - Let p>3 be a prime. For each maximal subgroup H⩽GL(d,p) with |H|⩾p3d+1, we construct a d-generator finite p-group G with the property that Aut(G) induces H on the Frattini quotient G/Φ(G) and |G|⩽p[Formula presented]. A significant feature of this construction is that |G| is very small compared to |H|, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on G/Φ(G), the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.

AB - Let p>3 be a prime. For each maximal subgroup H⩽GL(d,p) with |H|⩾p3d+1, we construct a d-generator finite p-group G with the property that Aut(G) induces H on the Frattini quotient G/Φ(G) and |G|⩽p[Formula presented]. A significant feature of this construction is that |G| is very small compared to |H|, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on G/Φ(G), the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.

UR - http://www.scopus.com/inward/record.url?scp=85035009575&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2017.11.006

DO - 10.1016/j.jpaa.2017.11.006

M3 - Article

VL - 222

SP - 2931

EP - 2951

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 10

ER -