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Finite 2-groups of class 2 in which every product of four elements can be reordered. (English) Zbl 0698.20013
If 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.
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