Questions tagged [verbal-subgroups]

A verbal subgroup of a group $G$, generated by the set of words $A \subset F_\infty$ ($F_\infty$ is a free group of countable rank) is a subgroup $V_A(G) = \langle \{h(w): w \in A, h \text{ is a homomorphism from } F_\infty \text{ to }G \} \rangle$. A verbal subgroup is always a characteristic one. To be used with the tag [group-theory].

3
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25 views

Is there a formula for $[F_n : V_{\{x^3\}}(F_n)]$?

Suppose $F_n$ is a free group of rank $n$. It is a rather well known fact, that $b_3(n) = [F_n : V_{\{x^3\}}(F_n)]$ is finite for all $n \in \mathbb{N}$. Is there a some sort of formula for $b_3(n)$? ...
2
votes
1answer
33 views

Are upper quasiverbal and lower quasiverbal subgroups always the same subgroup?

Let’s define a group quasiword as an element of $F_\infty \times P(F_\infty)$. Suppose $Q \subset F_\infty \times P(F_\infty)$ is a set of quasiwords. Define a prevariety described by $Q$ as a class ...
2
votes
1answer
64 views

Do the quasiverbal subgroups always exist?

Let’s define a group quasiword as an element of $F_\infty \times P(F_\infty)$. Suppose $Q \subset F_\infty \times P(F_\infty)$ is a set of quasiwords. Define a quasivariety described by $Q$ as a class ...
3
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0answers
52 views

Do all maximal verbal series have the same length?

Suppose, $G$ is a finite group. Let’s call a series of subgroups of $G$ $\{H_k\}_{k=1}^n$ a verbal series of length $n$, iff $H_1 = E$, $H_n = G$ and $\forall k < n$ $H_k$ is a verbal subgroup of $...
5
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0answers
52 views

Does there exist an Artinian verbally simple group, which is not characteristically simple?

Does there exist an Artinian verbally simple group, which is not characteristically simple? A characteristically simple group is a group without non-trivial proper characteristic subgroups, a verbally ...
2
votes
1answer
40 views

Does there exist a verbally simple group, which is not characteristically simple?

Does there exist a verbally simple group, which is not characteristically simple? A characteristically simple group is a group without non-trivial proper characteristic subgroups, a verbally simple ...
6
votes
1answer
88 views

Large counterexamples to “Non-isomorphic finite groups have verbal subgroups of different order”

In this question, it was conjectured that for every pair of non-isomorphic finite groups $G$ and $H$, there exists some word $\omega$ such that $|V_{\omega}(G)|\ne|V_{\omega}(H)|$, i.e. their ...
4
votes
1answer
67 views

Does $D_4$ have a verbal subgroup of order 4?

Does $D_4$ have a verbal subgroup of order 4? How did this question arise: In the comments $Q_8$ ad $D_4$ were pointed to be a possible counterexample to this question: Is it true, that for any two ...
5
votes
1answer
109 views

Does there exist some sort of classification of finite verbally simple groups?

Let’s call a group verbally simple if it does not have any non-trivial verbal subgroup. Does there exist some sort of classification of finite verbally simple groups? $G^n$, with $G$ being a finite ...
7
votes
1answer
182 views

Is it true, that for any two non-isomorphic finite groups $G$ and $H$ there exists such a group word $w$, that $|V_w(G)| \neq |V_w(H)|$?

Is it true, that for any two non-isomorphic finite groups $G$ and $H$ there exists such a group word $w$, that $|V_w(G)| \neq |V_w(H)|$? Here $V_w(G)$ stands for the verbal subgroup of $H$, generated ...
4
votes
0answers
121 views

Free groups: normal supplements of the commutator subgroup

Let $F$ be a free group and let $V$ be another verbal subgroup of $F$ such that $$ F = [F,F] V. $$ Is it true that $V=F?$ More generally, if $N$ is a normal (or even characteristic) subgroup of $F$ ...
0
votes
0answers
50 views

Alternating subgroup of symmetric group--is it considered a generalization of verbal?

The alternating subgroup ($A_n$) of the symmetric group($S_n$), is, I believe, not a verbal subgroup. But it is generated by elements of the form $xy$, where $x$ and $y$ have the same cycle structure. ...
2
votes
0answers
63 views

On a n-Engel Verbal Subgroup be perfect

Given a Group $G$, the $n$-th Engel Word defined by $[x,\ _0{x}]=x: \ [x,_{ \ n}y]=[[x,_{\ {n-1}}y],y]$ in the group $G$ consists by substituting group elements for the determinates. The Group ...
4
votes
2answers
89 views

When Verbal Subgroups are propers

Let $w$ be a group-word, and let $G$ be a group. The verbal subgroup $w(G)$ of $G$ determined by $w$ is the subgroup generated by the set consisting of values $w(g_1, \ldots, g_n)$, where $g_1, \ldots,...