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 ${\mathcal {S}}$:

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

Equivalence of definitions

Further information: equivalence of definitions of universal congruence condition

Relation with other properties

Weaker properties

• Collection of groups satisfying a strong normal replacement condition
• Collection of groups satisfying a weak normal replacement condition

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	$0\leq k\leq 5$	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	$0\leq k\leq 5$	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	$0\leq d\leq k\leq 5$	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 $2^{k}$	$p=2$	$k\geq 2$	Follows from above
Elementary abelian group of order $p^{k}$	all $p$	$k\geq 6$
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 $p$	$k\geq 6$	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 $p$	$k\geq 6$	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	$p=2$	$p=3$	$p=5$	$p=7$	$p\geq 11$
Abelian groups of order $p^{k}$	between 4 and 5	5	5	5	5
Abelian groups of order $p^{k}$, exponent dividing $p^{d}$, $2\leq d\leq k$	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 $p-1$