Congruence condition on number of subgroups of given prime power order
This article describes a congruence condition on an enumeration, or a count. It says that in a finite group and modulo prime number, the number of subgroups of given prime power order satisfies a congruence condition.
View other congruence conditions | View divisor relations
Statement
Version for a group of prime power order
Let be a group of prime power order and suppose . The following are true:
- The number of subgroups of of order is congruent to modulo .
- The number of normal subgroups of of order is congruent to 1 mod .
- The number of p-core-automorphism-invariant subgroups of of order is congruent to 1 mod .
In particular, the collection of groups of order is a collection of groups satisfying a universal congruence condition.
Version for a general finite group
Let be a finite group and be a prime power dividing the order of . Then, the number of subgroups of of order is congruent to 1 mod .
Related facts
Stronger facts
Corollaries
- Congruence condition on number of subgroups of given prime power order satisfying any given property weaker than normality
- Congruence condition on number of normal subgroups with quotient in a specific variety in a group of prime power order
Opposite facts
- Congruence condition fails for number of normal subgroups of given prime power order
- Congruence condition fails for number of characteristic subgroups in group of prime power order
- Congruence condition fails for number of central factors in group of prime power order
- Congruence condition fails for number of transitively normal subgroups in group of prime power order
- Congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group: In an abelian group of prime power order, the number of subgroups of a given order and bounded exponent is either zero or congruent to one modulo .
- Jonah-Konvisser abelian-to-normal replacement theorem: Jonah and Konvisser establish a congruence condition on the number of abelian subgroups of order for odd primes and small values of .
- Jonah-Konvisser elementary abelian-to-normal replacement theorem: Jonah and Konvisser establish a congruence condition on the number of elementary abelian subgroups of order for odd primes and small values of .
- Equivalence of definitions of universal congruence condition
- Congruence condition fails for subgroups of given prime power order and bounded exponent
Facts used
- Congruence condition relating number of subgroups in maximal subgroups and number of subgroups in the whole group
- Formula for number of maximal subgroups of group of prime power order
- Congruence condition relating number of normal subgroups containing minimal normal subgroups and number of normal subgroups in the whole group
- Formula for number of minimal normal subgroups of group of prime power order
Proof
Equivalence between the multiple formulations
For a proof of the equivalence of the three formulations for a group of prime power order and the formulation for a general finite group, see collection of groups satisfying a universal congruence condition and equivalence of definitions of universal congruence condition.
Proof for a group of prime power order in terms of all subgroups using Fact (1)
This proof uses the principle of mathematical induction in a nontrivial way (i.e., it would be hard to write the proof clearly without explicitly using induction).
This proof uses fact (1): congruence condition relating number of subgroups in maximal subgroups and number of subgroups in the whole group, combined with an induction on order.
Given: A group of order .
To prove: For any , the number of subgroups of order is congruent to modulo .
Proof: In this proof, we induct on , i.e., we assume the statement is true inside groups of order .
Base case for induction: The case is obvious.
Inductive step: If , the number of subgroups is 1, so the statement is true. So we consider .
For a subgroup of , denote by the number of subgroups of of order .
Step no. | Assertion/construction | Facts used | Given data/assumptions used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | . Here, means that is a maximal subgroup of . | Fact (1) | [SHOW MORE] | ||
2 | For every , . | inductive hypothesis | |||
3 | The number of maximal subgroups of is congruent to 1 mod . | Fact (2) | Fact-direct. | ||
4 | Steps (1)-(3) | [SHOW MORE] |
Proof for a group of prime power order using Fact (2)
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]References
Journal references
- A contribution to the theory of groups of prime power order by Philip Hall, Proceedings of the London Mathematical Society, ISSN 1460244X (online), ISSN 00246115 (print), Page 29 - 95(Year 1934): In this paper, Philip Hall summarizes and extends many of the known results about groups of prime power order.^{}^{More info}, Theorem 1.51: Hall proves the statement only for -groups; his proof is similar to the proof presented on the wiki page.