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
- Fourth isomorphism theorem
- 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 Facts (1) and (2)
The main advantage of this proof is that it has analogues for the congruence condition on number of subrings of given prime power order in nilpotent Lie ring.
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 , a prime number.
To prove: For any , the number of subgroups in 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 | [SHOW MORE] | ||
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 in terms of normal subgroups using Facts (3)-(5)
The main advantage of this proof is that it has analogues for the congruence condition on number of ideals of given prime power order in nilpotent Lie ring.
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 (3), combined with an induction on the order.
Given: A group of order , a prime number.
To prove: For any , the number of normal subgroups in 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 normal subgroups is 1, so the statement is true. So we consider .
For a subgroup of , denote by the number of normal subgroups of of order containing . Denote by the total number of normal subgroups of of order .
Step no. | Assertion/construction | Facts used | Given data/assumptions used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | where varies over all the minimal normal subgroups of . | Fact (3) | [SHOW MORE] | ||
2 | For each minimal normal subgroup of , is congruent to 1 mod . | Fact (4) | inductive hypothesis | [SHOW MORE] | |
3 | The number of minimal normal subgroups of is congruent to 1 mod . | Fact (5) | Fact-direct. | ||
4 | . | Steps (1), (2), (3) | [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.