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On the table of marks of a direct product of finite groups. (English) Zbl 1415.20007
Let \(G\) be a finite group. Remind that the Burnside ring \(\mathbf{B}(G)\) of \(G\) is the Grothendieck group of the category of finite \(G\)-sets, where addition and multiplication are induced by disjoint union and direct product, respectively. As an abelian group, it is free of rank \(n\), the number of conjugacy classes of subgroups of \(G\). The classes of the transitive \(G\)-sets \(G/H_1,\dots G/H_n\), where \(H_1,\dots,H_n\) are non-conjugate subgroups of \(G\), is a \(\mathbb{Z}\)-basis of \(\mathbf{B}(G)\). The table of marks of \(G\) is the \(n\times n\) matrix with integer coefficients (which is uniquely-defined up to conjugation by a permutation matrix) whose \((i,j)\)-entry is the cardinal of the set \[ \mathrm{Hom}_{G\text{-sets}}(G/H_i,G/H_j)\simeq (G/H_j)^{H_i}. \] These notions were introduced by W. Burnside [Theory of groups of finite order. Second edition. Cambridge: University Press (1911; JFM 42.0151.02)] more than 100 years ago to encode the combinatorics of the poset of conjugacy classes of subgroups of \(G\) (ordered by inclusion), which becomes quickly (with the size of \(G\)) very hard to understand. In [Exp. Math. 6, No. 3, 247–270 (1997; Zbl 0895.20017)], the second author gives a computational method to determine inductively the table of marks of finite groups.
The article under review introduces tools to compute the table of marks of a direct product of finite groups \(G_1\) and \(G_2\). The starting point is the classical lemma of E. Goursat [Ann. Sci. Éc. Norm. Supér. (3) 6, 9–102 (1889; JFM 21.0530.01)] giving an explicit bijection between the set of subgroups of \(G_1\times G_2\) and the set of pairs \((P_i,K_i)\) of subgroups of \(G_i\) (\(i\in\{1,2\}\)) such that \(K_i\triangleleft P_i\) endowed with a group isomorphism \(P_1/K_1\simeq P_2/K_2\). The authors define and study several posets (often with a group action) to describe the table of marks of \(G_1\times G_2\) from easier data – see Theorems 6.1, 6.2, 6.5 and 6.8. In the last section, they apply this method to the product of two copies of the symmetric group \(\mathfrak{S}_3\).

20C40 Computational methods (representations of groups) (MSC2010)
19A22 Frobenius induction, Burnside and representation rings
20D30 Series and lattices of subgroups
20D40 Products of subgroups of abstract finite groups
Full Text: DOI
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