TRANSVERSALS AS GENERATING SETS IN FINITELY GENERATED GROUPS
2015 ◽
Vol 93
(1)
◽
pp. 47-60
Keyword(s):
We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank-$n$ group $G$ and $H$ has index at least $n$ in $G$, we can construct a left transversal for $H$ which contains a generating set of size $n$ for $G$; this construction is algorithmic when $G$ is finitely presented. We also show that, in the case where $G$ has rank $n\leq 3$, there is a simultaneous left–right transversal for $H$ which contains a generating set of size $n$ for $G$. We finish by showing that if $H$ is a subgroup of a rank-$n$ group $G$ with index less than $3\cdot 2^{n-1}$, and $H$ contains no primitive elements of $G$, then $H$ is normal in $G$ and $G/H\cong C_{2}^{n}$.
2012 ◽
Vol 22
(05)
◽
pp. 1250048
◽
Keyword(s):
2005 ◽
Vol 37
(06)
◽
pp. 873-877
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2012 ◽
Vol 22
(03)
◽
pp. 1250026
Keyword(s):
2014 ◽
Vol 24
(05)
◽
pp. 609-653
◽
Keyword(s):
2011 ◽
Vol 21
(04)
◽
pp. 547-574
◽
Keyword(s):
1992 ◽
Vol 45
(3)
◽
pp. 513-520
◽