Sylow subgroup of holomorph of cyclic group of odd prime-cube order

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Definition

Let p be an odd prime number. This group is defined as the p-Sylow subgroup of the holomorph of the cyclic group of order p^3. Equivalently, it is the semidirect product of the cyclic group of order p^3 and the cyclic group of order p^2, where the generator of the latter acts on the former by the (p+1)-power map.

It can be given by the explicit presentation:

G := \langle a,b \mid a^{p^3} = b^{p^2} = e, bab^{-1} = a^{p + 1}\rangle

Note that the case p = 2 is different, because the automorphism structure of the cyclic group of order 2^3 is different, with a non-cyclic 2-Sylow subgroup of automorphisms. See holomorph of Z8 for details on that group.

Arithmetic functions

Function Value Explanation
order p^5 semidirect product of groups of order p^3 and p^2
exponent p^3 element a has order p^3, no element of order p^4
derived length 2 commutator subgroup \langle a^p \rangle cyclic of order p^2
nilpotency class 3
Frattini length 3 Frattini subgroup is \langle a^p, b^p, isomorphic to semidirect product of cyclic group of prime-square order and cyclic group of prime order
Fitting length 1 nilpotent.
minimum size of generating set 2 a,b
subgroup rank 2
rank as p-group 2 abelian subgroup \langle a^{p^2},b^p \rangle
normal rank 2 abelian normal subgroup \langle a^{p^2},b^p \rangle
characteristic rank 2 first omega subgroup is \langle a^{p^2},b^p \rangle

Group properties

Property Satisfied Explanation
abelian group No
group of prime power order Yes
metabelian group Yes
group of nilpotency class two No
metacyclic group Yes
finite p-group that is not characteristic in any finite p-group properly containing it Yes
regular p-group Yes for p \ge 5
Lazard Lie group Yes for p \ge 5

Subgroup-defining functions

Subgroup-defining function Value in presentation notation Isomorphism class Comment
center \langle a^{p^2} \rangle group of prime order
commutator subgroup \langle a^p \rangle cyclic group of prime-square order
Frattini subgroup \langle a^p, b^p \rangle semidirect product of cyclic group of prime-square order and cyclic group of prime order
second member of upper central series \langle a^p, b^p \rangle semidirect product of cyclic group of prime-square order and cyclic group of prime order
first omega subgroup \langle a^{p^2},b^p \rangle elementary abelian group of prime-square order
second omega subgroup \langle a^p, b \rangle
first agemo subgroup \langle a^p, b^p \rangle elementary abelian group of prime-square order
second agemo subgroup \langle a^{p^2} \rangle group of prime order

Facts

GAP implementation

Here, p</math> is the previously assigned value of the prime number, that we assume here to be odd. You can replace <tt>p by an actual numerical value of a prime or precede these commands by an assignment p := the value.

gap> C := CyclicGroup(p^3);
<pc group of size 343 with 3 generators>
gap> A := AutomorphismGroup(C);
<group of size 294 with 6 generators>
gap> S := SylowSubgroup(A,p);
<group>
gap> G := SemidirectProduct(S,C);
<pc group with 5 generators>