TY - JOUR
T1 - On Infinite Camina Groups
AU - Herzog, Marcel
AU - Longobardi, Patrizia
AU - Maj, Mercede
N1 - Funding Information:
The first author is grateful to the Department of Mathematics and Informatics of the University of Salerno for its hospitality and support, while this investigation was carried out.
PY - 2011/11
Y1 - 2011/11
N2 - A group G is called a Camina group if G′ ≠ G and each element x ∈ G\G′ satisfies the equation x G = xG′, where x G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).
AB - A group G is called a Camina group if G′ ≠ G and each element x ∈ G\G′ satisfies the equation x G = xG′, where x G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).
KW - Camina groups
KW - Centralizers
KW - Cosets
KW - Periodic groups
KW - Residually finite groups
UR - https://www.scopus.com/pages/publications/84857986185
U2 - 10.1080/00927872.2010.524684
DO - 10.1080/00927872.2010.524684
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AN - SCOPUS:84857986185
SN - 0092-7872
VL - 39
SP - 4403
EP - 4419
JO - Communications in Algebra
JF - Communications in Algebra
IS - 11
ER -