Orders of commutators and Products of conjugacy classes in finite groups

This paper establishes that a commutator $[x,g]$ is a $p$-element for all $g \in G$ if and only if $x$ is central modulo $\mathbf{O}_p(G)$, a result that generalizes the Baer--Suzuki and Glauberman $\mathbf{Z}_p^*$-theorems and is applied to prove that a conjugacy class $K$ satisfying $K^{-1}K = 1 \cup D \cup D^{-1}$ generates a solvable subgroup.

The Gist

Imagine a massive, chaotic dance party called The Finite Group. In this party, everyone is an element, and they move around according to strict rules. The paper you're asking about is like a detective story trying to figure out the "personality" of a specific dancer (let's call him X) based on how he interacts with everyone else.

Here is the breakdown of the paper's main ideas, translated into everyday language with some fun analogies.

The Main Characters

• The Group ($G$): The entire dance floor.
• The Element ($x$): A specific dancer we are watching.
• The Commutator ($[x, g]$): This is the "friction" or "mismatch" that happens when dancer $x$ tries to dance with dancer $g$.
  • If they dance perfectly in sync, the friction is zero (the result is 1).
  • If they clash, the friction has a specific "size" or "order" (how many steps it takes to get back to the start).
• $p$-element: A dancer whose "friction" size is a power of a specific prime number (like 2, 3, 5, etc.). Think of this as a dancer who only causes trouble in multiples of 3.

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The Big Discovery (Theorem 1.1)

The Question: What happens if dancer $x$ only causes friction that is a multiple of a specific number $p$ (a $p$-element) with every single person on the dance floor?

The Old Belief: Mathematicians had partial answers. They knew that if $x$ was a "prime-order" dancer, or if the friction was always 1 or a multiple of $p$, then $x$ was probably a "good citizen" (central) or part of a specific "safe zone" (a normal subgroup).

The New Discovery: The author proves a universal rule:

If $x$ only causes "p-friction" with everyone, then $x$ is essentially a "VIP" who doesn't really move relative to the "Safe Zone" ($O_p(G)$).

The Analogy: Imagine the dance floor has a "Safe Zone" (a VIP lounge) where everyone is allowed to be a bit chaotic. The theorem says: If your interactions with everyone else only ever result in chaos that fits inside the VIP lounge's rules, then you are effectively sitting still in that lounge. You aren't causing trouble outside the rules.

This unifies two famous previous theories (Baer-Suzuki and Glauberman) into one super-theory. It's like finding a single key that opens three different locks.

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The "Almost Simple" Challenge (Theorem 1.2)

To prove the big rule above, the author had to look at the most chaotic, complex dance floors possible: Almost Simple Groups. These are groups where the core is a "Simple Group" (a group that can't be broken down further, like a pure, indivisible atom of chaos).

The Claim: In these ultra-chaotic groups, if you pick any non-zero dancer $x$, there is always someone else on the floor ($g$) who will clash with $x$ in a way that creates friction not divisible by $p$.

The Analogy: It's like saying, "In a room full of pure chaos, you cannot find a person who only bumps into others in a specific, limited way. Eventually, someone will bump into you in a totally different, unpredictable way."

The author proved this by checking every type of "chaotic room" (using computer programs and complex math charts) and showing that no matter who you pick, you can't hide your "friction" forever.

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The Product of Conjugacy Classes (Theorem 1.4)

This part of the paper looks at Conjugacy Classes.

• Conjugacy Class ($K$): A group of dancers who are "twins." They might look different, but they have the exact same dance moves relative to the rest of the party.
• The Product ($K^{-1}K$): If you take every dancer in group $K$, reverse their move, and then pair them up with every other dancer in group $K$, what do you get?

The Scenario: The paper investigates a specific, rare situation:

If you mix group $K$ with its reverse, and the result is only the "peaceful center" (1) plus two other specific groups of twins ($D$ and $D^{-1}$), then the whole group of dancers generated by $K$ is Solvable.

What is "Solvable"? In math, a "solvable" group is one that can be broken down into simple, orderly layers. It's not a chaotic mess; it has a structure you can untangle.

The Analogy: Imagine you have a messy pile of tangled headphones ($K$). You shake them, and they only fall into three neat piles: the knot in the middle, and two identical tangles on the sides. The theorem says: "If the mess simplifies into just these three neat piles, then the headphones were never actually a tangled mess to begin with. They were just a simple, orderly knot."

This confirms a long-standing guess by other mathematicians: if a conjugacy class behaves this neatly, the group it generates is safe and orderly.

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The "Double Trouble" Rule (Theorem 1.5)

Finally, the paper looks at a very strict condition.

• Imagine dancer $x$ is a "p-dancer."
• The rule says: Every time $x$ clashes with someone, the friction is either zero OR the friction size is divisible by two specific different primes (say, 3 and 5). So, the friction must be at least 15, 30, 45, etc.

The Result: If this happens, $x$ must be Central.
The Analogy: If your only interactions with others are either "perfectly silent" or "explosively loud" (divisible by two different numbers), you are actually the King of the Party. You are so central that you don't actually move relative to anyone. You are the center of the universe for that group.

The cool thing about this proof is that it doesn't rely on the "Classification of Finite Simple Groups" (which is like a massive encyclopedia of every possible dance move in existence). The author found a clever, direct way to prove it without needing the whole encyclopedia.

Summary

This paper is a masterclass in finding order in chaos.

1. Main Rule: If your interactions are always "p-type," you are part of the "p-safe zone."
2. Chaos Check: In the most chaotic groups, you can't hide your interactions; someone will always clash with you in a new way.
3. Neatness Check: If mixing a group of twins only creates a few neat piles, the whole group is actually simple and orderly.
4. Strictness Check: If your interactions are either silent or "double-prime" loud, you are the center of the group.

The author, Hung P. Tong-Viet, has successfully connected several complex mathematical puzzles into a clearer picture, showing that even in the wildest mathematical groups, there are strict laws governing how elements interact.

Technical Summary

Here is a detailed technical summary of the paper "Orders of Commutators and Products of Conjugacy Classes in Finite Groups" by Hung P. Tong-Viet.

1. Problem Statement

The paper addresses two interconnected problems in finite group theory:

1. Characterization of Elements via Commutator Orders: Under what conditions does an element $x \in G$ lie in a specific characteristic subgroup (specifically, being central modulo the largest normal $p$-subgroup $O_p(G)$) based on the order properties of its commutators $[x, g]$?
2. Structure of Groups via Conjugacy Class Products: What can be deduced about the solvability of a subgroup generated by a conjugacy class $K$ if the product set $K^{-1}K$ has a restricted structure (specifically, $K^{-1}K = 1 \cup D \cup D^{-1}$ for some conjugacy class $D$)?

These problems generalize classical results like the Baer–Suzuki theorem and Glauberman's $Z^*_p$-theorem, aiming to unify them and resolve open conjectures regarding the decomposition of products of conjugacy classes.

2. Methodology

The author employs a combination of classical finite group theory, character theory, and the Classification of Finite Simple Groups (CFSG).

• Reduction to Almost Simple Groups: The proofs for the main theorems rely on a standard reduction strategy. The author assumes a minimal counterexample and reduces the problem to the case where the group $G$ is almost simple (i.e., $S \unlhd G \le \text{Aut}(S)$ for a non-abelian simple group $S$).
• Character Theory and Structure Constants: For the almost simple case, the author utilizes the complex group algebra $\mathbb{C}G$. Specifically, the paper uses:
  • Class Sums: Representing conjugacy classes as elements in the center of the group algebra.
  • Structure Constants: Calculating the coefficients in the product of class sums ($K^{-1}K = \sum n_i C_i$) using the formula involving irreducible characters: $n(K^{-1}, K, C_i) = \frac{|K|^2}{|G|} \sum_{\chi \in \text{Irr}(G)} \frac{|\chi(x)|^2 \chi(y_i)}{\chi(1)}$.
  • Defect Zero Characters: Leveraging the existence of irreducible characters of $p$-defect zero in groups of Lie type to force specific commutators to vanish or have specific orders.
• Case Analysis on Simple Groups:
  • Alternating Groups: Handled via direct calculation of cycle structures and supports of permutations.
  • Sporadic and Lie Type Groups: Handled using character tables (via GAP) and Deligne-Lusztig theory.
• Induction and Solvability Arguments: For the product of conjugacy classes, the proof uses induction on the group order, analyzing the solvable radical $R(G)$ and the generalized Fitting subgroup $F^*(G)$.

3. Key Contributions and Results

Theorem 1.1: Generalization of Baer–Suzuki and Glauberman

• Statement: Let $G$ be a finite group, $x \in G$, and $p$ a prime. The commutator $[x, g]$ is a $p$-element for every $g \in G$ if and only if $x$ is central modulo $O_p(G)$.
• Significance:
  • This unifies two major theorems:
    1. If $x$ is a $p'$-element (order not divisible by $p$), it recovers a result by Guralnick and Robinson (Theorem D(ii) in [21]).
    2. If $x$ is a $p$-element, it recovers the result by Guralnick and Malle (Theorem 1.4 in [18]), which states $x \in O_p(G)$.
  • It provides a common framework for determining when an element is "central" relative to a normal $p$-subgroup based solely on the $p$-power nature of its commutators.

Theorem 1.2: Commutators in Almost Simple Groups

• Statement: Let $G$ be a finite almost simple group with socle $S$, $x \in G$ non-trivial, and $p$ a prime. There exists $g \in G$ such that $[x, g]$ is a non-trivial $p'$-element.
• Role: This is the crucial technical lemma used to prove Theorem 1.1. It asserts that in an almost simple group, one cannot have all commutators of a non-trivial element being $p$-elements (or trivial). The proof involves a detailed case-by-case analysis of simple groups (Alternating, Sporadic, Lie Type) using character theory.

Theorem 1.4: Solvability of Subgroups from Restricted Products

• Statement: Let $K$ be a conjugacy class of $G$. If $K^{-1}K = 1 \cup D \cup D^{-1}$ for some conjugacy class $D$, then the subgroup $\langle K \rangle$ generated by $K$ is solvable.
• Significance:
  • This confirms a conjecture proposed in [6] in full generality.
  • Previous results required specific conditions (e.g., $D=D^{-1}$ or size constraints). This theorem removes those restrictions.
  • It implies that if the product of a conjugacy class with its inverse decomposes into at most three components (identity, a class, and its inverse), the group generated cannot be a non-abelian simple group.

Theorem 1.5: Central Elements via Commutator Orders (CFSG-free)

• Statement: Let $x$ be a $p$-element. If there exist distinct primes $r, s$ such that for all $g \in G$, either $[x, g] = 1$ or $rs$ divides the order of $[x, g]$, then $x \in Z(G)$.
• Significance:
  • This result is proven without using the Classification of Finite Simple Groups.
  • It can be viewed as a special case of Conjecture 2 in [22].
  • It demonstrates that if commutators are "large" (divisible by two distinct primes) or trivial, the element must be central.

4. Significance and Impact

• Unification: The paper successfully bridges the gap between the Baer–Suzuki theorem (concerning nilpotency and $p$-elements) and Glauberman's $Z^*_p$-theorem (concerning $p'$-elements and centralizers). Theorem 1.1 serves as a powerful generalization of both.
• Resolution of Conjectures: Theorem 1.4 settles a long-standing conjecture regarding the structure of groups where the product of a conjugacy class and its inverse is "small" (decomposes into few classes). This contributes to the broader understanding of the "product of conjugacy classes" problem, which is related to the Thompson conjecture and the Arad-Herzog conjecture.
• Methodological Rigor: The paper demonstrates the necessity of the Classification of Finite Simple Groups for the general case (Theorem 1.1 and 1.2) while also showing that specific variations (Theorem 1.5) can be resolved using purely structural group theory (Schur-Zassenhaus, Fitting subgroups).
• Applications: The results provide new criteria for identifying central elements and solvable subgroups, which are fundamental in the structural analysis of finite groups.

In summary, Tong-Viet's work advances the structural theory of finite groups by establishing precise conditions under which commutator orders dictate the position of an element within the group's normal structure, and by proving that specific constraints on conjugacy class products force the generated subgroups to be solvable.