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In mathematics and group theory, a block system for the action of a group ''G'' on a set ''X'' is a partition of ''X'' that is ''G''-invariant. In terms of the associated equivalence relation on ''X'', ''G''-invariance means that :''x'' ~ ''y'' implies ''gx'' ~ ''gy'' for all ''g'' in ''G'' and all ''x'', ''y'' in ''X''. The action of ''G'' on ''X'' determines a natural action of ''G'' on any block system for ''X''. Each element of the block system is called a block. A block can be characterized as a subset ''B'' of ''X'' such that for all ''g'' in ''G'', either *''gB'' = ''B'' (''g'' fixes ''B'') or *''gB'' ∩ ''B'' = ∅ (''g'' moves ''B'' entirely). If ''B'' is a block then ''gB'' is a block for any ''g'' in ''G''. If ''G'' acts transitively on ''X'', then the set is a block system on ''X''. The trivial partitions into singleton sets and the partition into one set ''X'' itself are block systems. A transitive ''G''-set ''X'' is said to be primitive if contains no nontrivial partitions. ==Stabilizers of blocks== If ''B'' is a block, the stabilizer of ''B'' is the subgroup :''G''''B'' = . The stabilizer of a block contains the stabilizer ''G''''x'' of each of its elements. Conversely, if ''x'' ∈ ''X'' and ''H'' is a subgroup of ''G'' containing ''G''''x'', then the orbit of ''x'' under ''H'' is a block. It follows that the blocks containing ''x'' are in one-to-one correspondence with the subgroups of ''G'' containing ''G''''x''. In particular, a ''G''-set is primitive if and only if the stabilizer of each point is a maximal subgroup of ''G''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Block (permutation group theory)」の詳細全文を読む スポンサード リンク
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