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Register a new userThe character tables of centralizers in Sporadic Simple Groups of ${rm HS}$ and ${rm Co_3}$
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To classify the finite dimensional pointed Hopf algebras with $G= {rm HS}$ or ${rm Co3}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {rm GAP}.
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To classify the finite dimensional pointed Hopf algebras with $G= {rm McL}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {rm GAP}.
We introduce and investigate a class of profinite groups defined via extensions of centralizers analogous to the extensively studied class of finitely generated fully residually free groups, that is, limit groups (in the sense of Z. Sela). From the fact that the profinite completion of limit groups belong to this class, results on their group-theoretical structure and homological properties are obtained.
The article deals with profinite groups in which the centralizers are pronilpotent (CN-groups). It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CN-group, and let F be the maximal normal pronilpotent subgroup of G. It is shown that F is open and the structure of the finite quotient G/F is described in detail.
The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CA-group. It is shown that G has a normal open subgroup N which is either abelian or pro-p. Further, a rather detailed information about the finite quotient G/N is obtained.
A group $G$ is said to have restricted centralizers if for each $g$ in $G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a set of primes $pi$, we take interest in profinite groups with restricted centralizers of $pi$-elements. It is shown that such a profinite group has an open subgroup of the form $Ptimes Q$, where $P$ is an abelian pro-$pi$ subgroup and $Q$ is a pro-$pi$ subgroup. This significantly strengthens a result from our earlier paper.
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