Order has only two prime factors implies solvable

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group whose order has at most two prime factors) must also satisfy the second group property (i.e., solvable group)
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Statement

The Burnside's p^aq^b theorem states that a group whose order has at most two prime factors (viz., a group whose order is of the form $p^aq^b$ where $p,q$ are primes and $a,b$ are nonnegative integers), must be solvable.

This includes the case of the trivial group as well as of a group of prime power order, though much stronger statements can be made in these cases.

Related facts

Order has only one prime factor

• Prime power order implies not centerless
• Prime power order implies nilpotent

Similar conditions for solvability

• Hall's theorem: This states that if $\pi$-Hall subgroups exist for all prime sets $\pi$, then the group is a solvable group.
• Odd-order implies solvable: Also known as the odd-order theorem and as the Feit-Thompson theorem, this states that any group of odd order is solvable.
• Order has only two prime factors implies prime divisor with larger prime power is core-nontrivial except in finitely many cases: This result is also called Burnside's other $p^aq^b$-theorem.
• Every Sylow subgroup is cyclic implies metacyclic
• Square-free implies solvability-forcing

More on order with few prime factors

• Classification of finite simple groups whose order has at most five prime factors counting multiplicities: These are precisely the projective special linear groups for primes $p$ where $p \in \{ 5,7,11,13\}$.
• Neumann's open problem on number of projective special linear groups whose order has exactly six prime factors counting multiplicities

More on groups whose order has only two prime factors

• Order has only two prime factors implies prime divisor with larger class two subgroups is core-nontrivial

Facts used

The table below lists key facts used directly and explicitly in the proof. Fact numbers as used in the table may be referenced in the proof. This table need not list facts used indirectly, i.e., facts that are used to prove these facts, and it need not list facts used implicitly through assumptions embedded in the choice of terminology and language.

Fact no.	Statement	Steps in the proof where it is used	Qualitative description of how it is used in the proof	What does it rely on?	Other applications
1	Sylow subgroups exist	Step (1) of first part of proof	Constructs a $p$-Sylow subgroup, which is then shown to be contained in the centralizer of a non-identity element.	Basic group theory, group actions	click here
2	Prime power order implies not centerless	Step (1) of first part of proof	Find a central element in a $p$-Sylow subgroup, which must therefore have a fairly big centralizer in the whole group.	Basic group theory, class equation of a group	click here
3	Simple abelian implies cyclic of prime order	Step (2) of first part of proof	Takes care of the case of nontrivial center	Basic group theory	click here
4	Conjugacy class of prime power size implies not simple	Step (3) of first part of proof	This fact itself is the hardest part of the proof, though its application to the proof is straightforward.	Character theory; hardest part of proof	click here
5	Lagrange's theorem	Step (1) of first part of proof, inductive step of second part of proof	Reducing from a group to a normal subgroup	Basic group theory	click here
6	Order of quotient group divides order of group	Inductive step of second part of proof	Reducing from a group to a quotient group	Basic group theory	click here
7	Solvability is extension-closed	Inductive step of second part of proof	Putting together solvability of normal subgroup and quotient group to get solvability of whole group	Basic definition of solvability	click here

Proof

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

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).

First part of proof: a nontrivial group whose order has only two prime factors is not simple unless it is cyclic of prime order

Given: A nontrivial group $G$ whose order is $p^aq^b$ for primes $p,q$ and nonnegative integers $a,b$.

To prove: $G$ is either cyclic of prime order or it is not simple.

Proof: Without loss of generality, assume that $a > 0$ (if both $a = b = 0$, $G$ is trivial and there is nothing to prove).

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	Either the center of $G$ is nontrivial, or $G$ has a conjugacy class of prime power order	Facts (1), (2), (5)	$|G| = p^aq^b$		[SHOW MORE] By Sylow's theorem (fact (1)), $G$ has a $p$-Sylow subgroup $P$. Since $P$ is a nontrivial $p$-group, fact (2) yields that the center $Z(P)$ is a nontrivial subgroup. Let $g$ be a non-identity element of $Z(P)$. Then $C_G(g)$ contains $P$, so the index $[G:C_G(g)]$, which is also equal to the size of the conjugacy class of $g$ in $G$, is a divisor of $q^b$ (this can be deduced from Fact (5), Lagrange's theorem: we get $P \le C_G(g)$, so the order of $C_G(g)$ is divisible by $p^a$, hence the index divides $p^aq^b/p^a = q^b$). In particular, either $g \in Z(G)$ or the size of the conjugacy class of $g$ is a nontrivial power of $q$. Thus, either the center is nontrivial or there is a conjugacy class of prime power order.
2	If the center of $G$ is nontrivial, then $G$ is either cyclic of prime order or not simple.	Fact (3)			[SHOW MORE] If $G$ has nontrivial center, then the center of $G$ is a nontrivial normal subgroup. This forces the center of $G$ to equal $G$. This, in turn, means that $G$ is a simple abelian group. Hence, by fact (3), $G$ must be a cyclic group of prime order.
3	If $G$ has a conjugacy class of prime power order, then $G$ is not simple.	Fact (4)			Fact-direct.
4	$G$ is either cyclic of prime order or not simple.			Steps (1),(2),(3)	Step-combination-direct.

Second part of proof: using induction

We prove the statement by a strong form of induction on the order. Specifically, the statement we are trying to prove is:

Statement to be proved by induction on $n$: If $n$ is of the form $p^aq^b$ where $p,q$ are primes and $a,b$ are nonnegative integers, then any group of order $n$ is solvable. Note that this includes the values of $n$ that are powers of only a single prime and it also includes $n = 1$, because $a,b$ are allowed to be zero.

Our strong form of induction uses the truth of the statement for all smaller orders. In other words:

Inductive hypothesis: The statement to be proved by induction is true for all $m < n$. In particular, it is true for all proper divisors of $n$.

Inductive goal: If $n = p^aq^b$ with $p,q$ primes and $a,b$ nonnegative integers, and $G$ is a group of order $n$, then $G$ is solvable.

Proof: We make two cases:

1. Case that $G$ is simple: In this case, the first part of proof yields that $G$ is cyclic of prime order, hence abelian, and hence solvable.
2. Case that $G$ is not simple: In this case, there is a proper nontrivial normal subgroup $N$ of $G$. By facts (5) and (6), the orders of $N$ and $G/N$ both divide the order of $G$, and hence both have at most two prime factors. Hence, the induction hypothesis applies to both (since $N$ is proper and nontrivial), and we obtain that $N$ and $G/N$ are both solvable. Thus, by fact (7), $G$ is solvable.

References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 886, Section 19.2 (Theorems of Burnside and Hall), (The proof of the theorem spans over pages 886-890, covering several definitions and lemmas.)^{More info}

Journal references

• On groups of order $p^\alpha q^\beta$ by William Burnside, Proceedings of the London Mathematical Society, ISSN 1460244X (online), ISSN 00246115 (print), Volume 1, Page 388 - 392(Year 1904): ^{More info}