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Concept
Product of group subsets
Résumé
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
The subsets 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. The product of two subgroups S and T of a group G is itself a subgroup of G if and only if ST = TS.
If S and T are subgroups of G, their product need not be a subgroup (for example, two distinct subgroups of order 2 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, 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. If both S and T are normal, then the product is normal as well.
If S and T are finite subgroups of a group 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.
The following modular law (for groups) holds for any Q a subgroup of S, where T is any other arbitrary subgroup (and both S and T are subgroups of some group G):
Q(S ∩ T) = S ∩ (QT).
The two products that appear in this equality are not necessarily subgroups.
If QT is a subgroup (equivalently, as noted above, if Q and T permute) then QT = ⟨Q ∪ T⟩ = Q ∨ T; i.e., QT is the join of Q and T in the lattice of subgroups of G, and the modular law for such a pair may also be written as Q ∨ (S ∩ T) = S ∩ (Q ∨ T), which is the equation that defines a modular lattice if it holds for any three elements of the lattice with Q ≤ S.
Source officielle
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