Collection of groups satisfying a weak normal replacement condition

From Groupprops
Jump to: navigation, search

Statement

Suppose \mathcal{S} is a collection of finite p-groups, i.e., groups of prime power order where the underlying prime is p. We say that \mathcal{S} satisfies a weak normal replacement condition if whenever a finite p-group P contains a subgroup isomorphic to an element of \mathcal{S}, P also contains a normal subgroup isomorphic to an element of \mathcal{S}.

Relation with other properties

Stronger properties

Examples/facts

Satisfaction

Collection of groups of order p^k Condition on prime p Condition on k Proof
Elementary abelian group of order p^k Odd 0 \le k \le 5 Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime
Elementary abelian group of order p^k Odd k \le (p + 5)/4 Elementary abelian-to-normal replacement theorem for large primes
Abelian groups of order p^k Odd 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 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^k, exponent dividing p^d Any 0 \le d \le k \le (p + 1)/2 Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one
Groups of order p^k, exponent p -- k < p Mann's replacement theorem for subgroups of prime exponent

Dissatisfaction

Collection of groups of order p^k Condition on prime p Condition 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 p^k p \ge 7 k \ge (p + 9)/2 elementary abelian-to-normal replacement fails for half of prime plus nine for prime greater than five
Abelian groups of order p^k p \ge 7 k \ge (p + 9)/2 abelian-to-normal replacement fails for half of prime plus nine for prime greater than five
Groups of order p^p, exponent p all p k = 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 weak normal replacement condition. The nature of all these is such that the weak normal replacement condition is satisfied for all smaller k but for no larger k. The between a and b below means that the minimum known value is a and the maximum known value is b.

Collection of groups p = 2 p = 3 p = 5 p = 7 p = 11 p \ge 11
Abelian groups of order p^k between 4 and 5 between 5 and 13 between 5 and 6 between 5 and 7 between 6 and 9 between (p + 1)/2 and (p + 7)/2
Abelian groups of order p^k, exponent dividing p^d, 2 \le d \le k between 2 and 5 between 5 and 13 between 5 and 6 between 5 and 7 between 6 and 9 between (p + 1)/2 and (p + 7)/2
Elementary abelian group of order p^k 1 between 5 and 13 (?) between 5 and 6 (?) between 5 and 7 between 6 and 9 between (p + 1)/2 and (p + 7)/2
Groups of exponent p, order p^k 1 at least 2 at least 4 at least 6