Achyuth Jayadevan

A finite modular lattice not embeddable in the lattice of formations of finite groups

We give a negative answer to Kourovka Problem 13.51 by constructing a modular lattice with 184 elements that admits no injective map preserving binary meets and joins into the lattice of formations of finite groups. The counterexample is the dual of an explicit incidence lattice...

A finite modular lattice not embeddable in the lattice of formations of finite groups

We give a negative answer to Kourovka Problem 13.51 by constructing a modular lattice with 184 elements that admits no injective map preserving binary meets and joins into the lattice of formations of finite groups. The counterexample is the dual of an explicit incidence lattice...

A nonstandard representation of PE₃(ℤ) by rational permutation matrices

Reduction modulo two and the action on the seven nonzero vectors of $\mathbb F_2^3$ define a homomorphism $PE_3(\mathbb Z)\longrightarrow\operatorname{GL}_7(\mathbb Q)$, where $PE_3(\mathbb Z)$ is the image of the elementary group in the ambient projective general linear group. W...

Centralizers of maximal order and nilpotent normal subgroups

We give a negative answer to Kourovka Problem 14.67. We construct a finite group $G$ of order $660602880$, an involution $a\in G$, and a nilpotent normal subgroup $H$ of order $16384$ such that $|C_G(a)|=41287680\ge |C_G(g)|$ for every $g\ne1$, but $H\nleq C_G(a)$. The group $H$...

Normalized root words and reflective power-subgroup completion

Fix a prime $p$, and let $P_n(G)$ be the subgroup generated by the $p^n$-th powers in a group $G$. We consider the full category of groups admitting a two-variable word $w$ and an integer $r\geq1$ such that $w(a,b)^p=a^p b^{p^r}$ and $w(a,1)=a$. Every nilpotent group belongs to t...

A nonstandard representation of PE₃(ℤ) by rational permutation matrices

Reduction modulo two and the action on the seven nonzero vectors of $\mathbb F_2^3$ define a homomorphism $PE_3(\mathbb Z)\longrightarrow\operatorname{GL}_7(\mathbb Q)$, where $PE_3(\mathbb Z)$ is the image of the elementary group in the ambient projective general linear group. W...

Normalized root words and reflective power-subgroup completion

Fix a prime $p$, and let $P_n(G)$ be the subgroup generated by the $p^n$-th powers in a group $G$. We consider the full category of groups admitting a two-variable word $w$ and an integer $r\geq1$ such that $w(a,b)^p=a^p b^{p^r}$ and $w(a,1)=a$. Every nilpotent group belongs to t...

Centralizers of maximal order and nilpotent normal subgroups

We give a negative answer to Kourovka Problem 14.67. We construct a finite group $G$ of order $660602880$, an involution $a\in G$, and a nilpotent normal subgroup $H$ of order $16384$ such that $|C_G(a)|=41287680\ge |C_G(g)|$ for every $g\ne1$, but $H\nleq C_G(a)$. The group $H$...

Computable two-variable laws for finite groups with abelian Sylow subgroups

For each prime $p$ we construct a computable sequence $a_{p,n}$ of words in two variables such that, for every finite group $G$ and every $n\geq |G|$, the identity $a_{p,n}=1$ holds in $G$ if and only if the Sylow $p$-subgroups of $G$ are abelian. When this condition fails, one f...

Compact families of noncompact subgroups and clopen partial orders

For a locally compact Hausdorff group \(G\), let \(\mathcal N(G)\) be the space of closed noncompact subgroups with the full Vietoris topology. We prove that a compact Hausdorff space \(X\) embeds in some \(\mathcal N(G)\) if and only if \(X\) admits a partial order with clopen p...

Compact families of noncompact subgroups and clopen partial orders

For a locally compact Hausdorff group \(G\), let \(\mathcal N(G)\) be the space of closed noncompact subgroups with the full Vietoris topology. We prove that a compact Hausdorff space \(X\) embeds in some \(\mathcal N(G)\) if and only if \(X\) admits a partial order with clopen p...

Identities of two-generated metabelian groups and finite conditions on Laurent ideals

Let $M$ be the free metabelian group of rank two and let $R=\mathbb{Z}[X^{\pm1},Y^{\pm1}]$. We classify the varieties generated by two-generated metabelian groups by pairs $(n,I)$, where $n\ge0$ and $I$ is an ideal of $R$. Admissibility is expressed by four power conditions and t...

Bounded-index nilpotent subgroups of rational linear groups with finitely many automorphism orbits

Let \(G\leq\operatorname{GL}_n(\mathbb Q)\) have finitely many orbits under its full abstract automorphism group, and put \(d=\dim_{\mathbb Q}\operatorname{span}_{\mathbb Q}G\). We prove that \(G\) has a torsion-free normal subgroup \(U\) satisfying \[ \gamma_n(U)=1,\qquad [G:U]\...

Nonunique indecomposable projective decompositions over the localized group ring of PSL2(F11)

Put \(O=\mathbb Z_{(11)}\) and \(G=\operatorname{PSL}_2(\mathbb F_{11})\). We construct nonzero finitely generated indecomposable projective \(O[G]\)-modules \(A,B,C,D\), each of \(O\)-rank \(44\), with \[A\oplus D\cong B\oplus C,\qquad A\not\cong B,\qquad A\not\cong C.\] Thus a...

An involutory automorphism of a double cover of A8 with central fixed involutions

We construct a finite perfect group \(E\) with centre \(\{1,z\}\) and \(E/\langle z\rangle\cong A_8\), together with an automorphism \(\alpha\) of order two satisfying \[\{g\in C_E(\alpha):g^2=1\}=\{1,z\}.\] The group \(E\) contains a subgroup isomorphic to \(C_2\times C_2\), so...

Divisible subgroups as exact equalizers in torsion-free nilpotent groups

Let \(G\) be torsion-free nilpotent of class at most \(c\), and let \(H\leq G\) be divisible. We construct a torsion-free nilpotent group \(K\) of class at most \(c\) and embeddings \(f_0,f_1:G\hookrightarrow K\) whose equalizer is exactly \(H\). Consequently the dominion of \(H\...

An irreducible nonmonomial representation of degree eight in a semiabelian group of order 2592

We construct a semiabelian group \(G\) of order \(2592\) with an irreducible complex representation of degree eight which is not induced from a linear character of any subgroup. Write \(M=C_2^3\rtimes A_4\) as a signed permutation group and choose \(K\leq M\) with \(K\cong Q_8\rt...

Nonclosed powers of a Zariski-closed subset of GL₃ in characteristic five

We construct a Zariski-closed subset \(Y\) of \(\mathrm{GL}_3(\overline{\mathbb F_5(u)})\) containing the identity such that \(Y^m\) is not closed for every integer \(m\geq2\). The same diagonal matrix lies in \(\overline{Y^m}\setminus Y^m\) for all such \(m\). The construction u...

Nonclosed powers of a Zariski-closed subset of GL₃ in characteristic five

We construct a Zariski-closed subset \(Y\) of \(\mathrm{GL}_3(\overline{\mathbb F_5(u)})\) containing the identity such that \(Y^m\) is not closed for every integer \(m\geq2\). The same diagonal matrix lies in \(\overline{Y^m}\setminus Y^m\) for all such \(m\). The construction u...

Palindromic length in free groups through reflection length and noncrossing matchings

We express palindromic length in a finitely generated free group as the minimum of two reflection lengths in a universal Coxeter group. The identity converts optimal reflection factorizations into optimal palindromic factorizations. Combined with the classical cancellation-norm r...