## On the normalizers of some classes of subgroups in braid groups.(English. Russian original)Zbl 1078.20037

Math. Notes 74, No. 1, 18-29 (2003); translation from Mat. Zametki 74, No. 1, 19-31 (2003).
The authors describe the normalizers of some classes of subgroups in braid groups. In particular, they consider the subgroup $$\Pi=\langle\sigma^4_1,\sigma^4_2\rangle\times\langle\sigma_4\sigma_3 \sigma_2\sigma^2_1\sigma_2\sigma_3\sigma_4\rangle\subset R_5$$ ($$R_5$$ denotes the subgroup of colored 5-braids). It is known that $$\Pi$$ is the direct product of free groups of rank 2. The authors find the normalizer $$N_{R_5}(\Pi)$$ in $$R_5$$. They also show that if $$H=H_1\times H_2$$ is a finitely generated subgroup of $$\Pi$$, $$H_1$$ and $$H_2$$ are not cyclic, then $$N_{K_5}(H)\subset N_{R_5}(\Pi)$$.

### MSC:

 20F36 Braid groups; Artin groups 20E07 Subgroup theorems; subgroup growth
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