Dear all,


I came across the above group (for any fixed odd prime $p$) and would need to know if it's large or not. It seems like it shouldn't be...


Best wishes,
Elisabeth



Is $G=\left<b_1, b_2, b_3 | [b_i^p, b_j^p]=1, \forall i,j=1,2,3\right>$ large?
The group $G$ maps onto the free product $C\_p\*C\_p\*C\_p$ of three cyclic groups of order $p$
(just send each $b\_i^p$ to $1$). This free product is virtually free, as a free product of finite groups (by Kurosh theorem the kernel of the homomorphism onto $C\_p \times C\_p \times C\_p$ is free) and is not virtually cyclic. Hence it is large, and so $G$ is large as well. This works for any $p>1$.


