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In mathematics, one can define a product of group subsets in a natural way. If ''S'' and ''T'' are subsets of a group ''G'', then their product is the subset of ''G'' defined by : Note that ''S'' and ''T'' need not be subgroups for this product to be well defined. The associativity of this product follows from that of the group product. The product of group subsets therefore defines a natural monoid structure on the power set of ''G''. A lot more can be said in the case where ''S'' and ''T'' are subgroups. == Product of subgroups == If ''S'' and ''T'' are subgroups of ''G'' their product need not be a subgroup (consider, for example, two distinct subgroups of order two in the symmetric group on 3 symbols). This product is sometimes called the ''Frobenius product''. In general, the product of two subgroups ''S'' and ''T'' is a subgroup if and only if ''ST'' = ''TS'', and the two subgroups are said to permute. (Walter Ledermann has called this fact the ''Product Theorem'',〔Walter Ledermann, ''Introduction to Group Theory'', 1976, Longman, ISBN 0-582-44180-3, p. 52〕 but this name, just like "Frobenius product" is by no means standard.) In this case, ''ST'' is the group generated by ''S'' and ''T'', i.e. ''ST'' = ''TS'' = ⟨''S'' ∪ ''T''⟩. If either ''S'' or ''T'' is normal then the condition ''ST''=''TS'' is satisfied and the product is a subgroup.〔Nicholson, 2012, Theorem 5, p. 125〕 If both ''S'' and ''T'' are normal, then the product is normal as well.〔 If ''G'' is a finite group and ''S'' and ''T'' are subgroups of ''G'', then ''ST'' is a subset of ''G'' of size ''|ST|'' given by the ''product formula'': : Note that this applies even if neither ''S'' nor ''T'' is normal. 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「Product of group subsets」の詳細全文を読む スポンサード リンク
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