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Questions about the branch of algebra that deals with groups.

2 votes
0 answers
137 views

Term and theories about "relation-free" elements in a group?

For a group $G$, there are two elements a, b which are "relation-free", i.e., there is no nonempty, reduced word $W(X,Y)$ such that $W(a,b)=1$ in $G$. Is there any terminologies or theories for that? …
qkqh's user avatar
  • 347
4 votes
0 answers
208 views

Image of the mapping class group of surfaces into automorphism group?

Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the p …
qkqh's user avatar
  • 347
2 votes
1 answer
549 views

injectivity of pushout?

We have the following pushout diagram: $$\begin{array}{ccc} \langle X, Y \rangle & \xrightarrow{\alpha} & \mathbb{Z}_a \ast \mathbb{Z}_b \ast \mathbb{Z}_c \ast \mathbb{Z}_d \\ \downarrow \scriptsize{\ …
qkqh's user avatar
  • 347
2 votes
1 answer
500 views

Do commutator functor and intersection commute?

For two subgroups $A, B$ in $G$, $[A,A] \cap [B,B] = [A\cap B, A \cap B]$? At least, if $G$ is free, is the left contained in the right?
qkqh's user avatar
  • 347
2 votes
1 answer
502 views

Commutator with a generator of a free group

Let $F$ be a free group $\langle x_1,...x_m\rangle$. If $a\in F_2$ and $[a,x_1] \in F_n$ then $a\in F_{n-1}$. Here, $F_n$ is the $n$-th lower central series term with $F_2=[F:F]$. How can I prove t …
qkqh's user avatar
  • 347
2 votes
2 answers
1k views

quotient groups of the lower central series of a free group

I have a question about some quotient groups of the lower central series of a free group. When there's a free group $F = \langle x_1,\cdots, x_n, y_1, \cdots, y_m\rangle $, let $A$ be the subgroup g …
qkqh's user avatar
  • 347
1 vote
1 answer
293 views

When $[G_k,G_m] = G_{k+m}$?

Hello? I have a simple question about combinatorial group theory. For a group $G$, I saw $[G_k, G_m] \subset G_{k+m}$ and these two subgroups need not be equal. Then is there any known condition that …
qkqh's user avatar
  • 347
1 vote
1 answer
992 views

Any subgroup of f.g. free group with finite index contains a term of lower central series?

Hello? I have some questions in the group theory. I know that the intersection of the lower central series of a finitely generate free group is trivial. So I wonder whether every nontrivial subgroup o …
qkqh's user avatar
  • 347