Finite 2-groups of class 2 in which every product of four elements can be reordered.

*(English)*Zbl 0698.20013If n is an integer greater than 1, then a group G belongs to the class \(P_ n\) if every ordered product of n elements can be reordered in at least one way; in other words, to each n-tuple \((x_ 1,x_ 2,...,x_ n)\) of elements of G there corresponds a non-trivial element \(\sigma\) of the symmetric group \(\Sigma_ n\) such that
\[
x_ 1x_ 2...x_ n=x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)}.
\]
The union of the classes \(P_ n\), \(n\geq 2\), is denoted by P. It was shown by M. Curzio, the first author, D. J. S. Robinson and M. Maj [Arch. Math. 44, 385-389 (1985; Zbl 0544.20036)] that P consists precisely of the finite-by-abelian-by-finite groups.

Clearly \(P_ 2\) is the class of abelian groups, while \(G\in P_ 3\) if and only if \(| G'| \leq 2\) [M. Curzio, the first author and M. Maj, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 74, 136-142 (1983; Zbl 0528.20031)]. Graham Higman characterized finite groups of odd order in \(P_ 4\) and also proved that a group G with \(G'\cong V_ 4\) (the 4-group) always belongs to \(P_ 4\). Then by the first author and M. Maj [Arch. Math. 49, 273-276 (1987; Zbl 0607.20017)], improving a result of M. Bianchi, R. Brandl and A. Gillio Berta Mauri [Arch. Math. 48, 281-285 (1987; Zbl 0623.20022)], it was shown that all \(P_ 4\)-groups are metabelian. Finally by M. Maj and the second author [Non-nilpotent groups in which every product of four elements can be reordered, Can. J. Math. (to appear)] the non-nilpotent \(P_ 4\)-groups were classified and the nilpotent \(P_ 4\)-groups were shown to have class at most 4. The present work is a further contribution to the classification of \(P_ 4\)-groups: finite 2-groups of class 2 belonging to \(P_ 4\) are precisely determined. It is shown that if G is such a group in \(P_ 4\), then \(G'\) has exponent at most 4.

The main results are: Theorem A. Let G be a finite 2-group of class 2 with \(G'\) of exponent 4. Then \(G\in P_ 4\) if and only if \(G'\cong C_ 4\) and G has a subgroup B of index 2 with \(| B'| =2\). Theorem B. Let G be a finite 2-group of class 2 with \(G'\) of exponent 2. Then \(G\in P_ 4\) if and only if (i) G has an abelian subgroup of index 2 or (ii) \(| G'| \leq 4\), or (iii) \(| G'| =8\) and G/Z(G) can be generated by 3 elements, or (iv) \(| G'| =8\), G/Z(G) can be generated by 4 elements and G is not the product of two abelian subgroups.

Clearly \(P_ 2\) is the class of abelian groups, while \(G\in P_ 3\) if and only if \(| G'| \leq 2\) [M. Curzio, the first author and M. Maj, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 74, 136-142 (1983; Zbl 0528.20031)]. Graham Higman characterized finite groups of odd order in \(P_ 4\) and also proved that a group G with \(G'\cong V_ 4\) (the 4-group) always belongs to \(P_ 4\). Then by the first author and M. Maj [Arch. Math. 49, 273-276 (1987; Zbl 0607.20017)], improving a result of M. Bianchi, R. Brandl and A. Gillio Berta Mauri [Arch. Math. 48, 281-285 (1987; Zbl 0623.20022)], it was shown that all \(P_ 4\)-groups are metabelian. Finally by M. Maj and the second author [Non-nilpotent groups in which every product of four elements can be reordered, Can. J. Math. (to appear)] the non-nilpotent \(P_ 4\)-groups were classified and the nilpotent \(P_ 4\)-groups were shown to have class at most 4. The present work is a further contribution to the classification of \(P_ 4\)-groups: finite 2-groups of class 2 belonging to \(P_ 4\) are precisely determined. It is shown that if G is such a group in \(P_ 4\), then \(G'\) has exponent at most 4.

The main results are: Theorem A. Let G be a finite 2-group of class 2 with \(G'\) of exponent 4. Then \(G\in P_ 4\) if and only if \(G'\cong C_ 4\) and G has a subgroup B of index 2 with \(| B'| =2\). Theorem B. Let G be a finite 2-group of class 2 with \(G'\) of exponent 2. Then \(G\in P_ 4\) if and only if (i) G has an abelian subgroup of index 2 or (ii) \(| G'| \leq 4\), or (iii) \(| G'| =8\) and G/Z(G) can be generated by 3 elements, or (iv) \(| G'| =8\), G/Z(G) can be generated by 4 elements and G is not the product of two abelian subgroups.

Reviewer: P.Longobardi

##### MSC:

20D15 | Finite nilpotent groups, \(p\)-groups |

20D60 | Arithmetic and combinatorial problems involving abstract finite groups |

20F05 | Generators, relations, and presentations of groups |

20F12 | Commutator calculus |

20E10 | Quasivarieties and varieties of groups |

20F24 | FC-groups and their generalizations |