Collection of groups satisfying a universal congruence condition

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Definition

Suppose $\mathcal{S}$ is a finite collection of finite $p$ -groups, groups of prime power order for the prime $p$ . We say that $\mathcal{S}$ satisfies a universal congruence condition if the following equivalent conditions are satisfied by $\mathcal{S}$ :

For any finite $p$ -group $P$ that contains a subgroup isomorphic to an element of $\mathcal{S}$ , the number of subgroups of $P$ isomorphic to elements of $\mathcal{S}$ is congruent to $1$ modulo $p$ .
For any finite $p$ -group $P$ that contains a subgroup isomorphic to an element of $\mathcal{S}$ , the number of normal subgroups of $P$ isomorphic to elements of $\mathcal{S}$ is congruent to $1$ modulo $p$ .
For any finite $p$ -group $P$ and any normal subgroup $Q$ of $P$ such that $Q$ contains a subgroup isomorphic to an element of $\mathcal{S}$ , the number of normal subgroups of $P$ isomorphic to elements of $\mathcal{S}$ and contained in $Q$ is congruent to $1$ modulo $p$ .
For any finite $p$ -group $P$ that contains a subgroup isomorphic to an element of $\mathcal{S}$ , the number of p-core-automorphism-invariant subgroups of $P$ isomorphic to elements of $\mathcal{S}$ is congruent to $1$ modulo $P$ .
For any finite group $G$ containing a subgroup isomorphic to an element of $\mathcal{S}$ , the number of subgroups of $G$ isomorphic to an element of $\mathcal{S}$ is congruent to $1$ modulo $p$ .

Relation with other properties

Weaker properties

Examples/facts

Satisfaction

Collection	Conditions on prime $p$	Conditions on $k$	Proof
All groups of order $p k$	all $p$	all $k$	congruence condition on number of subgroups of given prime power order
Elementary abelian group of order $p k$	odd prime	$0 \le k \le 5$	Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime
Abelian groups of order $p k$	odd prime	$0 \le k \le 5$	Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
Abelian groups of order $p k$ , exponent dividing $p d$	odd prime	$0 \le d \le k \le 5$	Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime
Abelian groups of order $p 3$	all $p$	$k = 3$	congruence condition on number of abelian subgroups of prime-cube order
Abelian groups of order $p 4$	all $p$	$k = 4$	congruence condition on number of abelian subgroups of prime-fourth order
Abelian groups of order $8$ , exponent dividing $4$	$p = 2$	$k = 3$	congruence condition on number of abelian subgroups of order eight and exponent dividing four
Abelian groups of order $16$ , exponent dividing $8$	$p = 2$	$k = 4$	congruence condition on number of abelian subgroups of order sixteen and exponent dividing eight
Non-cyclic groups of order $p 3$	odd $p$	$k = 3$	congruence condition on number of non-cyclic subgroups of prime-cube order for odd prime

Dissatisfaction

Collection	Conditions on prime $p$	Conditions on $k$	Proof
Klein four-group	$p = 2$	$k = 2$	elementary abelian-to-normal replacement fails for Klein four-group
Elementary abelian group of order $2 k$	$p = 2$	$k \ge 2$	Follows from above
Elementary abelian group of order $p k$	all $p$	$k \ge 6$
Abelian groups of order $p 6$	all $p$	$k = 6$	Congruence condition fails for abelian subgroups of prime-sixth order
Abelian groups of order $p k$	all $p$	$k \ge 6$	Follows from above.
Elementary abelian group of order $p 6$	all $p$	$k = 6$	Congruence condition fails for elementary abelian subgroups of prime-sixth order (example same as for abelian subgroups)
Elementary abelian group of order $p k$	all $p$	$k \ge 6$	Follows from above.
Groups of order $p p$ , exponent $p$	all $p$	$k = p$	Congruence condition fails for subgroups of order p^p and exponent p

Threshold values

This lists threshold values of $k$ : the largest value of $k$ for which the collection of $p$ -groups of order $p k$ satisfying the stated condition satisfies a universal congruence condition. The nature of all these is such that the universal congruence condition is satisfied for all smaller $k$ but for no larger $k$ . We use between $a$ and $b$ to mean that the value is at least $a$ and at most $b$ .

Collection of groups	$p = 2$	$p = 3$	$p = 5$	$p = 7$	$p \ge 11$
Abelian groups of order $p k$	between 4 and 5	5	5	5	5
Abelian groups of order $p k$ , exponent dividing $p d$ , $2 \le d \le k$	between 3 and 5	5	5	5	5
Elementary abelian group of order $p k$	1	5	5	5	5
Groups of exponent $p$ , order $p k$	1	2	between 2 and 4	between 2 and 6	between 2 and $p - 1$

Facts about Collection of groups satisfying a universal congruence conditionRDF feed

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Stronger than	Collection of groups satisfying a strong normal replacement condition +, and Collection of groups satisfying a weak normal replacement condition +

Collection of groups satisfying a universal congruence condition

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