Product of subgroups

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Definition

Symbol-free definition

The Product of two subgroups of a group is the subset consiting of the paarwise products between the two subgroups.

Definition with symbols

The Product HK of two subgroups H and K of a group G is:

HK := \{ hk \mid h \in H, k \in K \}.

If G is abelian and if the group operation is denoted as +, the product is termed the sum, and is denoted H + K:

H + K = \{ h + k \mid h \in H, k \in K \}.

Facts

HK is the double coset HeK, e being the identity element of G.

The cardinality \left| HK \right| of HK is \left| H \right| \left| K \right| / \left| H \cap K  \right|.

The product HK is in general not a subgroup, because it may not be closed under the group operation.

The smallest subgroup containing HK is the join \langle H, K \rangle of H and K, which is also the subgroup generated by H and K.

Following statements are equivalent:

  • HK is a subgroup
  • HK =  \langle H,K \rangle, viz., it is precisely the join of H and K (the subgroup generated by H and K)
  • \! HK = KH
  • HK \subseteq KH
  • KH \subseteq HK

If the above equivalent conditions hold, H and K are termed permuting subgroups.