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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 $$\mathfrak{S}_3$$.

##### MSC:
 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
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