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A family of Fitting classes of supersoluble groups. (English) Zbl 0839.20026

The aim of the present paper is to construct a Fitting class \(\mathfrak u\) that contains only supersoluble groups but not only nilpotent groups.
Let \(p\) be a prime, \(p\equiv 1\bmod 3\), \(n\) a primitive cube root of unity in \(\text{GF}(p)\) and \[ \begin{aligned} T&= \langle a,b\mid a^p= b^p= [a,b,a,a ]- [a,b,a,b ]= [a,b,b,b ]=1\rangle,\\ U&= \langle T,s\mid s^3=1,\;a^s= a^n,\;b^s= b^n \rangle.\end{aligned} \] The group \(U\) is supersoluble. Let \({\mathfrak u}_0\) be the class of all finite groups \(G= XY\), where \(X= O_p(G)\) and \(Y\in \text{Syl}_3 (G)\) such that \(X\) is a central product of groups \(T_i \cong T\) and \(Y/C_Y (T_i) \cong C_3\) and \(T_iY/ C_Y (T_i) \cong U\) for all indices \(i\). Let \({\mathfrak u}\) be the class of all groups \(G\in {\mathfrak S}_p {\mathfrak S}_3\). where \(O^p (G)\in {\mathfrak u}_0\).
In this paper it is proved that \(\mathfrak u\) is the Fitting class generated by \(U\) i.e. \(\mathfrak u=\text{Fit\,}U\). Like similar constructions the Fitting class \({\mathfrak u}\) can be described in terms of a more general pattern, the Fitting classes of Dark type. Namely, it is proved: Theorem. Let \(p\) be a prime, \(p\equiv 1\bmod 3\), \(n\) a primitive cube root of unity in \(\text{GF}(p)\), \(d\geq 2\). Set \[ \begin{aligned} T &= \langle a_1, \dots, a_d\mid a^p_1= \dots= a^p_d= [a_i, a_j, a_k, a_l]=1 \text{ for all } i,j,k,l\rangle\\ \text{and} U&=\langle T,s\mid s^3= a^s_1 a_1^{-n}= \dots= a^s_d a_d^{-n} =1\rangle. \end{aligned} \] Then \(\text{Fit\, }U\) is a supersoluble non-nilpotent Fitting class.

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20F17 Formations of groups, Fitting classes
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References:

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