Baer–Suzuki Theorem Overview

• The Baer–Suzuki theorem is a finite group theory criterion that detects membership in the largest normal p-subgroup using pairs of conjugates.
• It extends to π-radicals by analyzing bounded tuples of conjugates and setting sharp upper and lower bounds based on the smallest prime outside π.
• Recent work refines this theory with product, commutator, and solvable-radical variants, offering precise local generation conditions for both solvable and nonsolvable cases.

The Baer–Suzuki theorem is a criterion in finite group theory for detecting membership in the $p$-radical from conjugacy data. In its classical form, if $p$ is a prime, $G$ a finite group, and $x\in G$, then

$$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$

where $O_p(G)$ is the largest normal $p$-subgroup of $G$ (Yang et al., 2019). The theorem says that membership in the largest normal $p$-subgroup can be detected purely from the way an element interacts with its conjugates. Modern work has extended this philosophy to $\pi$-radicals, normal subsets, commutator and product conditions, and solvable-radical analogues, while also clarifying that the “Baer theorem” appearing in Lie superalgebra theory is a different theorem about central series rather than a Baer–Suzuki-type statement (Yang et al., 2021, Nayak, 2018).

1. Classical formulation and immediate interpretations

The standard finite-group formulation is the following:

Let $p$0 be a prime, $p$1 a finite group, and $p$2. Then $p$3 (Yang et al., 2019)

Here $p$4 is the conjugacy class of $p$5. In the language of conjugacy classes, the theorem gives a width-$p$6 criterion for the $p$7-radical: an element belongs to $p$8 exactly when every pair consisting of the element and one of its conjugates generates a $p$9-group (Yang et al., 2021).

For involutions, the product formulation becomes especially transparent. If $G$0 is a conjugacy class of involutions, then for $G$1,

$G$2

Thus, for $G$3, the Baer–Suzuki condition can be rephrased in terms of products (Parker et al., 2022).

The theorem is specific to the $G$4-radical. For a general set of primes $G$5, a naive two-conjugate analogue fails, and much of the later literature studies how many conjugates are actually needed, or which alternative local conditions still force membership in a radical subgroup (Yang et al., 2019).

2. From $G$6-radicals to $G$7-radicals

Let $G$8 be a set of primes. A finite group $G$9 is a $x\in G$0-subgroup if every prime divisor of $x\in G$1 lies in $x\in G$2, and $x\in G$3 is the largest normal $x\in G$4-subgroup of $x\in G$5 (Yang et al., 2021). The direct analogue of Baer–Suzuki with “pair of conjugates” and “$x\in G$6-subgroup” is false in general (Yang et al., 2019).

A standard counterexample uses symmetric groups. Given $x\in G$7, choose a prime $x\in G$8 and a set $x\in G$9 such that $$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$0, $$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$1 contains all primes less than $$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$2, and $$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$3. Then in $$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$4, any $$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$5 transpositions generate a $$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$6-subgroup, but

$$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$7

So no fixed $$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$8 works for all sets $$x\in O_p(G)\quad\Longleftrightarrow\quad \langle x,x^g\rangle \text{ is a }p\text{-group for every }g\in G,$$9 if one only asks about $O_p(G)$0 conjugates when $O_p(G)$1 is too small (Yang et al., 2019).

To measure the correct width, one defines $O_p(G)$2 to be the least integer $O_p(G)$3 such that for every finite group $O_p(G)$4,

$O_p(G)$5

The existence theorem states that for every set of primes $O_p(G)$6, there exists a natural number $O_p(G)$7 depending on $O_p(G)$8 such that membership in $O_p(G)$9 is detected by every $p$0 conjugates (Yang et al., 2019).

If $p$1 is the smallest prime not in $p$2, then the general upper bound proved for proper $p$3 is

$p$4

The same line of work records the lower bound $p$5, so the sharp problem is to determine whether the optimal width is $p$6 for $p$7 and $p$8 for $p$9 (Yang et al., 2019, Yang et al., 2021).

3. Sharp Baer–Suzuki theorems for $G$0-radicals

The sharp $G$1-Baer–Suzuki conjecture takes the following form. Let $G$2 be the smallest prime not in $G$3, and define

$G$4

Then for a conjugacy class $G$5 of a finite group $G$6,

$G$7

This is the sharp width conjectured in the modern $G$8-radical program (Yang et al., 2021, Yang et al., 2021).

Setting	Sharp statement	Source
Proper $G$9	$p$0	(Yang et al., 2019)
Nonabelian composition factors alternating, linear, or unitary	Sharp width $p$1 for $p$2, $p$3 for $p$4	(Yang et al., 2021)
Nonabelian composition factors sporadic or alternating	Same sharp width	(Yang et al., 2021)

The proofs reduce to almost simple groups and introduce the invariants $p$5 and $p$6. For a nonabelian simple group $p$7 and $p$8, $p$9 is the smallest number of $\pi$0-conjugates of $\pi$1 needed to generate $\pi$2, while $\pi$3 is the smallest number of $\pi$4-conjugates of $\pi$5 needed to generate a subgroup whose order is divisible by $\pi$6 (Yang et al., 2021, Yang et al., 2021).

For finite simple linear and unitary groups $\pi$7 or $\pi$8, with $\pi$9 of prime order, the sharp local bound is

$p$00

where $p$01 is chosen as in the theorem. This yields the sharp $p$02-Baer–Suzuki theorem for groups whose nonabelian composition factors are alternating, linear, or unitary simple groups (Yang et al., 2021). The same bound is proved for the 26 sporadic simple groups, and hence for groups whose nonabelian composition factors are sporadic or alternating (Yang et al., 2021).

A key conceptual point is that the sharp theory no longer tests a single pair of conjugates. It tests bounded tuples of conjugates, and the optimal bound depends only on the smallest prime outside $p$03 (Yang et al., 2021).

4. Product, commutator, and normal-subset variants

One direction of the literature replaces the subgroup-generation hypothesis by a product condition. Let $p$04 be a prime, $p$05 a finite group, and $p$06 a non-empty normal subset of elements of order $p$07. If every element of

$p$08

is a $p$09-element, then for $p$10, the group $p$11 is soluble. Moreover, if $p$12, then $p$13 is odd, $p$14 is a non-trivial $p$15-group, and $p$16 is an elementary abelian $p$17-group (Parker et al., 2022). The alternating group $p$18, generated by a conjugacy class of $p$19-elements with $p$20 again a conjugacy class of $p$21-elements, shows that the conclusion cannot be strengthened from soluble to nilpotent in general (Parker et al., 2022).

A second direction studies normal subsets $p$22 of $p$23-elements. If for every $p$24, the subgroup $p$25 is a $p$26-group with no section isomorphic to $p$27, then

$p$28

This is a two-class variation of Baer–Suzuki. In the same framework, if $p$29 is a conjugacy class of $p$30-elements and $p$31 consists of $p$32-elements, then $p$33 (Guralnick et al., 2013).

The paper "Variations on the Baer--Suzuki Theorem" also studies commutator-closed normal subsets. If $p$34 is a normal subset of $p$35-elements and $p$36 is closed under taking commutators, then either $p$37 is a $p$38-group, or $p$39 and

$p$40

is a direct product of copies of $p$41, with $p$42 not closed under squares (Guralnick et al., 2013).

A third direction introduces weakly subnormal subgroups. A subgroup $p$43 is weakly subnormal in $p$44 if $p$45 is not subnormal in $p$46 but it is subnormal in every proper overgroup of $p$47 in $p$48. By Wielandt’s Zipper Lemma, a weakly subnormal subgroup $p$49 lies in a unique maximal subgroup $p$50 of $p$51, and if $p$52 is a $p$53-group then

$p$54

This framework yields several Baer–Suzuki-type criteria, including: $p$55

$p$56

and

$p$57

There is also the order-theoretic characterization

$p$58

(Guralnick et al., 2024).

5. Solvable analogues and nonsolvable generation

The Baer–Suzuki philosophy also has solvable-radical analogues. If $p$59 has prime order $p$60, then

$p$61

Equivalently, if $p$62 is not in the solvable radical, then there exists $p$63 such that $p$64 is not solvable (Guest, 2010).

For almost simple groups, the statements become sharper. If $p$65 is almost simple and $p$66 has prime order $p$67, then there exists an involution $p$68 such that

$p$69

If $p$70 has order $p$71 or $p$72, then there exists $p$73 such that

$p$74

(Guest, 2010).

Involutions require a three-conjugate formulation, since $p$75 is dihedral when $p$76 is an involution. The almost simple result states that either there exist $p$77 such that

$p$78

or $p$79 belongs to an explicit exception list. The listed exceptions include: transpositions and triple transpositions in $p$80; unitary transvections in $p$81; graph automorphisms in $p$82 and $p$83; orthogonal transvections in $p$84; symplectic transvections in $p$85; reflections in $p$86; and the classes $p$87, $p$88, and $p$89 in $p$90, $p$91, and $p$92 (Guest, 2010).

The order-$p$93 case is not valid in complete generality. The example

$p$94

where $p$95 is a transposition in $p$96 and $p$97 is a transvection in $p$98, has $p$99 of order $G$00, trivial solvable radical, and yet $G$01 is solvable for all $G$02 (Guest, 2010). This marks a genuine boundary of the solvable analogue.

6. Distinction from Baer’s theorem in Lie superalgebras

The term “Baer’s theorem” can refer to a different theorem. In the Lie superalgebra paper "On Isoclinism {data} Baer's theorem for Lie superalgebras," the theorem called “Baer’s theorem” is not Baer–Suzuki-type. It is a finiteness theorem relating upper central quotients and lower central terms (Nayak, 2018).

Specifically, if $G$03 is a Lie superalgebra and

$G$04

then

$G$05

For $G$06, this yields

$G$07

The same paper proves a converse-type result: if the commutator set consisting of elements of length $G$08 is finite, then $G$09 is finite-dimensional, and if $G$10 is generated by $G$11 elements, then

$G$12

(Nayak, 2018).

The surrounding structure theory in that paper concerns isoclinism, stem extensions, stem covers, and the Schur multiplier

$G$13

for a free presentation $G$14 (Nayak, 2018). Nothing there concerns conjugacy classes, $G$15-elements, or the local generation criteria characteristic of Baer–Suzuki theory. The connection is only historical in the sense that both bear Baer’s name; there is no Baer–Suzuki-type result discussed (Nayak, 2018).