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Strong conciseness of coprime commutators in profinite groups
Group Theory · Submitted 15 Mar 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim A profinite group $G$ is finite-by-pronilpotent if and only if there is $k$ such that the set of $\gamma_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$.
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Let $G$ be a profinite group. The coprime commutators $\gamma_j^*$ and $\delta_j^*$ are defined as follows. Every element of $G$ is both a $\gamma_1^*$-value and a $\delta_0^*$-value. For $j\geq 2$, let $X$ be the set of all elements of $G$ that are powers of $\gamma_{j-1}^*$-values. An element $a$ is a $\gamma_j^*$-value if there exist $x\in X$ and $g\in G$ such that $a=[x,g]$ and $(|x|,|g|)=1$. For $j\geq 1$, let $Y$ be the set of all elements of $G$ that are powers of $\delta_{j-1}^*$-values. The element $a$ is a $\delta_j^*$-value if there exist $x,y\in Y$ such that $a=[x,y]$ and $(|x|,|y|)=1$. In this paper we establish the following results. A profinite group $G$ is finite-by-pronilpotent if and only if there is $k$ such that the set of $\gamma_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$. A profinite group $G$ is finite-by-(prosoluble of Fitting height at most $k$) if and only if there is $k$ such that the set of $\delta_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $G$ be a profinite group. | background 1.00 | 0.04 | 0.01 | unclear 0.57 |
| s1 | The coprime commutators $\gamma_j^*$ and $\delta_j^*$ are defined as follows. | background 0.92 | 0.15 | 0.01 | unclear 0.56 |
| s2 | Every element of $G$ is both a $\gamma_1^*$-value and a $\delta_0^*$-value. | background 0.88 | 0.20 | 0.03 | asserted 1.00 |
| s3 | For $j\geq 2$, let $X$ be the set of all elements of $G$ that are powers of $\gamma_{j-1}^*$-values. | background 0.75 | 0.11 | 0.01 | asserted 0.67 |
| s4 | An element $a$ is a $\gamma_j^*$-value if there exist $x\in X$ and $g\in G$ such that $a=[x,g]$ and $(|x|,|g|)=1$. | background 0.91 | 0.11 | 0.02 | asserted 0.95 |
| s5 | For $j\geq 1$, let $Y$ be the set of all elements of $G$ that are powers of $\delta_{j-1}^*$-values. | background 0.72 | 0.15 | 0.01 | asserted 0.86 |
| s6 | The element $a$ is a $\delta_j^*$-value if there exist $x,y\in Y$ such that $a=[x,y]$ and $(|x|,|y|)=1$. | background 0.88 | 0.11 | 0.02 | asserted 0.96 |
| s7 | In this paper we establish the following results. | contribution 0.99 | 0.92 | 0.04 | asserted 0.79 |
| s8 | A profinite group $G$ is finite-by-pronilpotent if and only if there is $k$ such that the set of $\gamma_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$. | result 1.00 | 0.93 | 0.42 | asserted 1.00 |
| s9 | A profinite group $G$ is finite-by-(prosoluble of Fitting height at most $k$) if and only if there is $k$ such that the set of $\delta_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$. | result 1.00 | 0.90 | 0.37 | asserted 1.00 |
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Original abstract
Let $G$ be a profinite group. The coprime commutators $\gamma_j^*$ and $\delta_j^*$ are defined as follows. Every element of $G$ is both a $\gamma_1^*$-value and a $\delta_0^*$-value. For $j\geq 2$, let $X$ be the set of all elements of $G$ that are powers of $\gamma_{j-1}^*$-values. An element $a$ is a $\gamma_j^*$-value if there exist $x\in X$ and $g\in G$ such that $a=[x,g]$ and $(|x|,|g|)=1$. For $j\geq 1$, let $Y$ be the set of all elements of $G$ that are powers of $\delta_{j-1}^*$-values. The element $a$ is a $\delta_j^*$-value if there exist $x,y\in Y$ such that $a=[x,y]$ and $(|x|,|y|)=1$. In this paper we establish the following results. A profinite group $G$ is finite-by-pronilpotent if and only if there is $k$ such that the set of $\gamma_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$. A profinite group $G$ is finite-by-(prosoluble of Fitting height at most $k$) if and only if there is $k$ such that the set of $\delta_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$.
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Uses the method or tool of 1 paper
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We will use Lemma [ref] for $ϕ=γ_k^*$ or $ϕ=δ_k^*$, but it could be applied to other cases, such as group word maps or uniform (anti-coprime) commutators (see [ref] or [ref] ).
Commutators, centralizers, and strong conciseness in profinite groups · Coprime commutators in prosoluble groups · 0.81
Cites as background 1 paper
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A number of new results on strong conciseness of group words can be found in [ref] .
Strong conciseness of Engel words in profinite groups · Introduction · 0.96
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