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YCor
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Reduced Free Groupfree group

Let $G$ be a $d$-generated group. Then my first question is how to see free reduced group $FV(G)$ in the variety containing $G$. What I understood is: "Let $W \subset F_d$ (d$d$-generated free group) be the collection of words which are laws for $G$, i.e. for every $w \in W$ image of word map $w$ on $G$ is identity. Let $F_m$ be m-generated free group and $S$ be the union of images of words $w \in W$ in $F_m$. If $FV(G)$ is a mthe $m$-generated reduced free group then $FV(G) = \frac{F_m}{\langle S \rangle}$."

My second question is when $G$ is finite group then is it true that finitely generated free reduced group $FV(G)$ will also be finite? Where can I read more about free reduced group?

Thank you!

Reduced Free Group

Let $G$ be a $d$-generated group. Then my first question is how to see free reduced group $FV(G)$ in the variety containing $G$. What I understood is: "Let $W \subset F_d$ (d-generated free group) be collection of words which are laws for $G$, i.e. for every $w \in W$ image of word map $w$ on $G$ is identity. Let $F_m$ be m-generated free group and $S$ be the union of images of words $w \in W$ in $F_m$. If $FV(G)$ is a m-generated reduced free group then $FV(G) = \frac{F_m}{\langle S \rangle}$."

My second question is when $G$ is finite group then is it true that finitely generated free reduced group $FV(G)$ will also be finite? Where can I read more about free reduced group?

Thank you!

Reduced free group

Let $G$ be a $d$-generated group. Then my first question is how to see free reduced group $FV(G)$ in the variety containing $G$. What I understood is: "Let $W \subset F_d$ ($d$-generated free group) be the collection of words which are laws for $G$, i.e. for every $w \in W$ image of word map $w$ on $G$ is identity. Let $F_m$ be m-generated free group and $S$ be the union of images of words $w \in W$ in $F_m$. If $FV(G)$ is the $m$-generated reduced free group then $FV(G) = \frac{F_m}{\langle S \rangle}$."

My second question is when $G$ is finite group then is it true that finitely generated free reduced group $FV(G)$ will also be finite? Where can I read more about free reduced group?

Thank you!

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Shri
Shri
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Reduced Free Group

Let $G$ be a $d$-generated group. Then my first question is how to see free reduced group $FV(G)$ in the variety containing $G$. What I understood is: "Let $W \subset F_d$ (d-generated free group) be collection of words which are laws for $G$, i.e. for every $w \in W$ image of word map $w$ on $G$ is identity. Let $F_m$ be m-generated free group and $S$ be the union of images of words $w \in W$ in $F_m$. If $FV(G)$ is a m-generated reduced free group then $FV(G) = \frac{F_m}{\langle S \rangle}$."

My second question is when $G$ is finite group then is it true that finitely generated free reduced group $FV(G)$ will also be finite? Where can I read more about free reduced group?

Thank you!