Mutually Permutable Product in Finite Groups

• Mutually permutable product is a finite group factorization where each subgroup permutes with every subgroup of the other, enabling a rigorous subgroup calculus.
• It provides precise control over commutator subgroups and residuals, leading to clear criteria for supersolubility and nilpotency.
• The framework extends to msp-permutability, impacting generalized Fitting height, non-p-soluble length, and other formation properties.

Mutually permutable product denotes a factorization $G=AB$ of a finite group in which the two factors satisfy strong subgroup-permutability conditions. In the cited literature, one formal presentation requires $UB=BU$ for every subgroup $U\le A$ and $AV=VA$ for every subgroup $V\le B$; other presentations state that $A$ permutes with every subgroup of $B$ and $B$ permutes with every subgroup of $A$, and write this as $XY=YX$ for all $UB=BU$0 and $UB=BU$1 (Monakhov, 2016, Felipe et al., 2017, Murashka et al., 2023). Within finite-group theory, this hypothesis supports a substantial structure theory: it controls commutator subgroups and residuals, yields supersolubility criteria from local class-size conditions, admits a weaker prime-separated analogue called msp-permutability, and sharply constrains generalized Fitting height and non-$UB=BU$2-soluble length (Monakhov, 2016, Felipe et al., 2017, Monakhov et al., 2020, Murashka et al., 2023).

1. Formal definition and basic subgroup calculus

All of the cited papers work in the category of finite groups. Standard commutator notation is used throughout: for $UB=BU$3, $UB=BU$4; for subgroups $UB=BU$5, $UB=BU$6 is the subgroup generated by all such commutators; and the derived subgroup is $UB=BU$7 (Monakhov, 2016). Normal closure is written $UB=BU$8 for the smallest normal subgroup of $UB=BU$9 containing $U\le A$0 (Monakhov, 2016).

For mutually permutable products, several closure properties are basic. If $U\le A$1 is mutually permutable and $U\le A$2, then $U\le A$3 is again a mutually permutable product. If $U\le A$4, then $U\le A$5, and $U\le A$6 and $U\le A$7 are mutually permutable; if $U\le A$8, then $U\le A$9. In particular, if $AV=VA$0, then $AV=VA$1 and $AV=VA$2 are mutually permutable (Felipe et al., 2017). A further structural fact used repeatedly is that, for a nontrivial mutually permutable product, the product of the normal closures $AV=VA$3 is nontrivial; in practice this is used to ensure the existence of a minimal normal subgroup lying in one factor (Felipe et al., 2017).

The commutator calculus of a mutually permutable factorization is especially rigid. If $AV=VA$4, then $AV=VA$5; if $AV=VA$6, then $AV=VA$7; and

$AV=VA$8

These identities isolate the inter-factor commutator subgroup $AV=VA$9 as a canonical normal subgroup and make it possible to pass from global properties of $V\le B$0 to properties of $V\le B$1, $V\le B$2, $V\le B$3, and $V\le B$4 (Monakhov, 2016). In later work on generalized Fitting height, this structural picture is complemented by the facts that $V\le B$5 and $V\le B$6 are subnormal in $V\le B$7, and that minimal normal subgroups $V\le B$8 satisfy $V\le B$9 (Murashka et al., 2023).

2. Residual theory, supersolubility, and the commutator subgroup

Residual arguments in this area are formulated via formations. If $A$0 is a formation, then the $A$1-residual $A$2 is the smallest normal subgroup $A$3 such that $A$4. For the formation $A$5 of supersoluble groups, $A$6 is the supersoluble residual; for the formation $A$7 of nilpotent groups, $A$8 is the nilpotent residual (Monakhov, 2016). For finite groups, the nilpotent residual equals the last term of the lower central series: $A$9

A central theorem states that if $B$0 is the mutually permutable product of supersoluble subgroups $B$1 and $B$2, then

$B$3

Since the groups are finite, this is equivalently

$B$4

This identifies the supersoluble residual of the whole group with the nilpotent residual of the derived subgroup and, even more specifically, with the nilpotent residual of the inter-factor commutator subgroup (Monakhov, 2016).

The proof combines several standard tools. First, one shows $B$5 by passing to $B$6: there $B$7 and $B$8 are normal, and $B$9 is nilpotent because it arises from the images of $B$0 and $B$1, which are nilpotent when $B$2 and $B$3 are supersoluble; Baer’s theorem then yields supersolubility of $B$4. For the reverse containment, one considers $B$5; its derived subgroup is nilpotent by construction, the images of $B$6 and $B$7 remain supersoluble, mutual permutability passes to quotients, and known results on mutually permutable products of supersoluble groups imply that the quotient is supersoluble. Finally, since $B$8, the nilpotent normal closures of $B$9 and $A$0 contribute only nilpotent normal factors, so the nilpotent residual of $A$1 is exactly $A$2 (Monakhov, 2016).

Several corollaries follow immediately. The equality $A$3 gives the criterion

$A$4

In particular, if $A$5 is nilpotent, then $A$6 is supersoluble. The theorem also provides a computational reduction: instead of determining the supersoluble residual directly inside $A$7, one may compute the last term of the lower central series of $A$8 (Monakhov, 2016).

The same paper proves a prime-local analogue. If $A$9 and $XY=YX$0 are $XY=YX$1-supersoluble and $XY=YX$2 is mutually permutable, then

$XY=YX$3

so the $XY=YX$4-supersoluble residual of $XY=YX$5 equals the $XY=YX$6-nilpotent residual of $XY=YX$7 and of $XY=YX$8; consequently, if $XY=YX$9 is $UB=BU$00-nilpotent, then $UB=BU$01 is $UB=BU$02-supersoluble (Monakhov, 2016).

3. Conjugacy class sizes and structural restrictions

A different line of research studies mutually permutable products through class sizes. For $UB=BU$03, the conjugacy class size is $UB=BU$04. For a prime $UB=BU$05, its $UB=BU$06-part is

$UB=BU$07

The condition $UB=BU$08 is imposed on elements of $UB=BU$09, usually $UB=BU$10-regular elements of prime power order, and then translated into constraints on Sylow $UB=BU$11-subgroups and on coprime actions (Felipe et al., 2017).

For a fixed prime $UB=BU$12 with $UB=BU$13, if $UB=BU$14 for any $UB=BU$15-regular element $UB=BU$16 of prime power order, then $UB=BU$17 is soluble, $UB=BU$18 is $UB=BU$19-nilpotent, and the Sylow $UB=BU$20-subgroups of $UB=BU$21 are elementary abelian. If, under the same local class-size restriction, $UB=BU$22 is $UB=BU$23-soluble, then $UB=BU$24 is $UB=BU$25-supersoluble (Felipe et al., 2017). These results combine Hall subgroup factorization across the factors, location of minimal normal subgroups inside a factor, and control of $UB=BU$26-kernels under coprime action.

Global versions are stronger. If for every prime $UB=BU$27 and every prime power order $UB=BU$28-regular element $UB=BU$29, one has $UB=BU$30, then $UB=BU$31 is supersoluble and $UB=BU$32 has elementary abelian Sylow subgroups. If the hypothesis is strengthened to all $UB=BU$33-regular elements $UB=BU$34, then the order of a Sylow $UB=BU$35-subgroup of $UB=BU$36 is at most $UB=BU$37. Under the strongest assumption, namely that $UB=BU$38 is square-free for each $UB=BU$39, the conclusions become: $UB=BU$40 is supersoluble, $UB=BU$41 and $UB=BU$42 have elementary abelian Sylow subgroups, $UB=BU$43 is abelian, and for each prime $UB=BU$44, both $UB=BU$45 and $UB=BU$46 have Sylow $UB=BU$47-subgroups of order at most $UB=BU$48 (Felipe et al., 2017).

The proof machinery is adapted to mutual permutability. Hall $UB=BU$49-subgroups can be chosen in factorized form $UB=BU$50; minimal normal subgroups can be taken inside one factor; and a key coprime-action lemma states that if a $UB=BU$51-group $UB=BU$52 acts faithfully on an elementary abelian $UB=BU$53-group $UB=BU$54 and $UB=BU$55 for each $UB=BU$56, then $UB=BU$57 is cyclic. In the present setting, the class-size hypotheses give $UB=BU$58 and hence $UB=BU$59, which forces cyclic or elementary-abelian behavior in the relevant sections (Felipe et al., 2017).

4. The prime-separated variant: msp-permutability

A later development replaces full mutual permutability by a weaker prime-separated condition. The subgroups $UB=BU$60 and $UB=BU$61 of $UB=BU$62 are called msp-permutable if $UB=BU$63 and, whenever $UB=BU$64 is a Sylow $UB=BU$65-subgroup of $UB=BU$66 and $UB=BU$67 is a Sylow $UB=BU$68-subgroup of $UB=BU$69 with $UB=BU$70, the subgroups $UB=BU$71 and $UB=BU$72 are mutually permutable. The same paper recalls the classical terminology: $UB=BU$73 and $UB=BU$74 are mutually permutable if $UB=BU$75 and $UB=BU$76 for all $UB=BU$77, $UB=BU$78, and totally permutable if $UB=BU$79 for all such $UB=BU$80 and $UB=BU$81 (Monakhov et al., 2020).

The relation to the classical notion is explicit: msp-permutability weakens and specializes mutual permutability by requiring interaction only between Sylow subgroups of distinct primes and by building in the subgroup condition $UB=BU$82. Despite that weakening, several familiar closure properties survive. If $UB=BU$83 with $UB=BU$84 msp-permutable and $UB=BU$85, then $UB=BU$86 is again an msp-permutable product. If $UB=BU$87, then $UB=BU$88 and the two factors are msp-permutable in $UB=BU$89. In suitable $UB=BU$90-groups, Hall $UB=BU$91-subgroups also factor as $UB=BU$92 in msp-permutable form (Monakhov et al., 2020).

The structural consequences parallel mutually permutable theory. If $UB=BU$93 and $UB=BU$94 are soluble and $UB=BU$95 is an msp-permutable product, then $UB=BU$96 is soluble. If $UB=BU$97 is the largest prime in $UB=BU$98 and both $UB=BU$99 and $U\le A$00 are $U\le A$01-closed, then $U\le A$02 is $U\le A$03-closed; if $U\le A$04 is the smallest prime in $U\le A$05 and both factors are $U\le A$06-nilpotent, then $U\le A$07 is $U\le A$08-nilpotent; and if both factors have an ordered Sylow tower of supersoluble type, then so does $U\le A$09 (Monakhov et al., 2020).

At the formation level, if $U\le A$10 is a subgroup-closed saturated formation with

$U\le A$11

and $U\le A$12 is a product of msp-permutable subgroups $U\le A$13, then $U\le A$14. The most immediate corollary is the supersolubility statement: if $U\le A$15 and $U\le A$16 are supersoluble and msp-permutable, then $U\le A$17 is supersoluble. The same closure also holds for the formations $U\le A$18 of widely supersoluble groups and $U\le A$19 of groups whose primary cyclic subgroups are $U\le A$20-subnormal (Monakhov et al., 2020).

5. Generalized Fitting height, non-$U\le A$21-soluble length, and non-Frattini length

The mutually permutable hypothesis also controls non-soluble structure. For a finite group $U\le A$22, the generalized Fitting subgroup is

$U\le A$23

where $U\le A$24 is the Fitting subgroup and $U\le A$25 is the layer. Its iterates are defined by

$U\le A$26

and the generalized Fitting height $U\le A$27 is the least $U\le A$28 such that $U\le A$29. For a prime $U\le A$30, the non-$U\le A$31-soluble length $U\le A$32 is defined from the shortest normal series

$U\le A$33

with $U\le A$34-soluble factors in odd positions and non-empty direct products of nonabelian simple groups in even positions; then $U\le A$35 (Murashka et al., 2023).

If $U\le A$36 is a mutually permutable product, then

$U\le A$37

and for every prime $U\le A$38,

$U\le A$39

In particular, the nonsoluble length satisfies $U\le A$40. In the soluble case, since $U\le A$41, the height bound specializes to

$U\le A$42

recovering Jabara’s bound (Murashka et al., 2023).

The upper bound $U\le A$43 is sharp: $U\le A$44 is a mutually permutable product of its Sylow subgroups $U\le A$45 and $U\le A$46, with $U\le A$47 and $U\le A$48. A further special case states that if $U\le A$49 and $U\le A$50 are quasinilpotent, then $U\le A$51 (Murashka et al., 2023). The proofs are functorial: the paper develops a general theory of hereditary functorials, uses subnormal joins to deduce exact height formulas, and combines minimal normal subgroup analysis with formation-hypercenter arguments. The same work states that the arguments avoid reliance on the Classification of Finite Simple Groups (Murashka et al., 2023).

The non-Frattini length $U\le A$52 is defined from the shortest normal series

$U\le A$53

whose even factors lie in the Frattini subgroup of the quotient and whose odd factors are non-empty direct products of simple groups. It satisfies

$U\le A$54

and for each $U\le A$55 there exists a group $U\le A$56 with $U\le A$57 and $U\le A$58. For totally permutable products one has

$U\le A$59

Two open questions are recorded: whether the lower bound $U\le A$60 always holds for totally permutable products, and whether there exists a universal constant $U\le A$61 such that $U\le A$62 for mutually permutable products (Murashka et al., 2023).

6. Examples, scope, and limitations

Several examples illustrate how the theory behaves in concrete cases. If $U\le A$63 and $U\le A$64 are supersoluble and $U\le A$65, then the mutually permutable condition holds and $U\le A$66; with $U\le A$67 and $U\le A$68, the theorem on residuals gives $U\le A$69, so $U\le A$70 is supersoluble. For the dihedral group

$U\le A$71

taking $U\le A$72 and $U\le A$73 yields a mutually permutable product with $U\le A$74, and again $U\le A$75. More generally, if $U\le A$76 is a nonabelian $U\le A$77-group generated by subgroups $U\le A$78 and $U\le A$79 with $U\le A$80, then $U\le A$81 and $U\le A$82 are supersoluble, $U\le A$83, mutual permutability holds, and $U\le A$84 is a nontrivial $U\le A$85-group, hence nilpotent; the residual theorem again yields $U\le A$86 (Monakhov, 2016).

Class-size methods admit examples of a different kind. The group $U\le A$87 factorizes as a mutually permutable product $U\le A$88 and $U\le A$89 a Sylow $U\le A$90-subgroup; for $U\le A$91, the hypothesis of the $U\le A$92-supersolubility theorem holds on prime-power $U\le A$93-regular elements in $U\le A$94, and since $U\le A$95 is $U\le A$96-soluble, the theorem yields that $U\le A$97 is $U\le A$98-supersoluble. At the same time, the hypotheses have clear limits. There exists a group of order $U\le A$99 with GAP id $AV=VA$00 that factorizes as a mutually permutable product $AV=VA$01 and $AV=VA$02, such that $AV=VA$03 satisfies Theorem B for $AV=VA$04, but some $AV=VA$05 has $AV=VA$06 divisible by $AV=VA$07; this shows that the class-size hypothesis is imposed on elements of the factors as embedded in $AV=VA$08, and is not automatically inherited factor-internally unless the factor is normal. Another cautionary example shows that even when $AV=VA$09 is mutually permutable and $AV=VA$10 for all $AV=VA$11, the subgroup $AV=VA$12 need not be abelian: one may take $AV=VA$13 and $AV=VA$14. A further example, with $AV=VA$15 for distinct odd primes $AV=VA$16, shows that Theorem E can apply while $AV=VA$17 has a large Sylow $AV=VA$18-subgroup, so the stronger hypothesis of Theorem F is genuinely needed for the bound $AV=VA$19 (Felipe et al., 2017).

The scope of the theory is explicitly finite-group theoretic in all cited work. The residual equality

$AV=VA$20

is proved only under the hypotheses that $AV=VA$21 and $AV=VA$22 are supersoluble and $AV=VA$23 is mutually permutable; it is not claimed for arbitrary solvable factors (Monakhov, 2016). Likewise, the sharp bounds for generalized Fitting height and exact formulas for non-$AV=VA$24-soluble length are specific to mutually permutable products; for arbitrary factorizations $AV=VA$25, the cited literature records that there is no bound on $AV=VA$26 or on nonsoluble length in terms of the corresponding invariants of $AV=VA$27 and $AV=VA$28 alone (Murashka et al., 2023). These contrasts explain why mutually permutable products occupy a distinct position among factorized finite groups: the hypothesis is strong enough to force precise commutator, residual, and height-theoretic behavior, yet flexible enough to support several nontrivial variants and refinements across supersoluble and nonsoluble settings.