Finite groups with a trivial Chermak-Delgado subgroup. (English) Zbl 1436.20029
Let \(G\) be a finite group. The Chermak-Delgado lattice \(CD(G)\) of \(G\) is a modular sublattice of the lattice \(L(G)\) of subgroups of \(G\) consisting of all subgroups \(H\) with maximal Chermak-Delgado measure \(m_G(H)=|H||C_G(H)|\). The centralizer map \(X\rightarrow C_G(X)\) on \(L(G)\) induces a duality in \(CD(G)\). It follows that the least element of the lattice \(CD(G)\), the Chermak-Delgado subgroup of \(G\), is a characteristic, abelian subgroup that contains the center of \(G\). The author studies groups for which this subgroup is trivial, that is, for which \(1\in CD(G)\).
It is well-known that \(CD(G_1\times \dots \times G_n)=CD(G_1)\times \dots \times CD(G_n)\) for finite groups \(G_i\). The author proves that, conversely, if \(1\in CD(G)\) and \(CD(G)\simeq L_1\times \dots \times L_n\) for lattices \(L_i\), then \(G=G_1 \times \dots \times G_n\) with \(CD(G_i)\simeq L_i\) for all \(i\in \{ 1,\dots ,n\}\). He shows that the assumption \(1\in CD(G)\) is needed for this and he proves further properties of such groups: if \(1\in CD(G)\), then \(CD(G)\) neither contains nontrivial \(p\)-groups nor maximal subgroups of \(G\). On the other hand, he finally constructs examples of groups \(G\) with \(1\in CD(G)\) and \(CD(G)\) indecomposable such that \(CD(G)\not= \{ 1,G\}\).
It is well-known that \(CD(G_1\times \dots \times G_n)=CD(G_1)\times \dots \times CD(G_n)\) for finite groups \(G_i\). The author proves that, conversely, if \(1\in CD(G)\) and \(CD(G)\simeq L_1\times \dots \times L_n\) for lattices \(L_i\), then \(G=G_1 \times \dots \times G_n\) with \(CD(G_i)\simeq L_i\) for all \(i\in \{ 1,\dots ,n\}\). He shows that the assumption \(1\in CD(G)\) is needed for this and he proves further properties of such groups: if \(1\in CD(G)\), then \(CD(G)\) neither contains nontrivial \(p\)-groups nor maximal subgroups of \(G\). On the other hand, he finally constructs examples of groups \(G\) with \(1\in CD(G)\) and \(CD(G)\) indecomposable such that \(CD(G)\not= \{ 1,G\}\).
Reviewer: Roland Schmidt (Kiel)
References:
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