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2.45 (1966)

Partially Solved

(P. Hall). Prove or refute the following conjectures:
$\qquad$ a) If a word $v$ takes only finitely many values in a group $G$, then the verbal subgroup $vG$ is finite.
$\qquad$ b) If the marginal subgroup $v^*G$ has finite index $m$ in $G$, then the order of $vG$ is finite and divides a power of $m$.
$\qquad$ c) If $G$ satisfies the maximum condition and $vG$ is finite, then $v^*G$ has finite index in $G$.

Contributor: Yu. I. Merzlyakov

Progress

a), c) These conjectures are refuted in: a) (S. V. Ivanov, Soviet Math. (Izv. VUZ), 33, no. 6 (1989), 59–70); c) (I. S. Ashmanov, A. Yu. Olshanskii, Soviet Math. (Izv. VUZ), 29, no. 11 (1985), 65–82).

b) There are examples when $|vG|$ does not divide a power of $m$ (Yu. G. Kleiman, Trans. Moscow Math. Soc., 1983, no. 2, 63–110), but the question of finiteness remains open.

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