For the subgroup H of a groups G middot show that The rule t

For the subgroup H of a groups (G, middot) show that The rule that associates to a pair of h epsilon H and g epsilon G the element h * g h middot g is an action H on G. The rule that associates to a pair of h epsilon H and g epsilon G the element h * g g middot h^-1 is an action H on G. In the case when G = Z and H is the subgroup 3Z of multiples of 3, find all orbits of the action * of H on G. How many of them are there?

Solution

Since H is a subgroup of G, they both have the same identity element by definition. Let us denote the identity by \'e\'.

The group and the subgroup operation is denoted by : ( ^. )

The subgroup H is said to \"act\" on the group G, following a rule (*), if this rule satisfies the following two conditions :

i) e * g = g (where g is an element of G)

ii) h1 * (h2 * g) = (h1 ^. h2) * g (where h1, h2 are elements of H, g is an element of G)

(1) Here the action is defined as : h * g = h ^. g

Now condition i) is obviously satisfied since \'e\' is the identity of H as well G (H being the subgroup of G), so e * g = e ^. g = g ( the dot (^.) being the group operation, and by definition of identity element for any \'g\' belonging to G, e ^. g =g ). For condition ii) we have h2 * g = h2 ^. g and the element h2 ^. g belongs to the group G, by the closure property of the group G. So h1 * (h2 ^. g ) is well defined, and it is simplified to h1 ^. (h2 ^. g). RHS of condition ii) simplifies after using the definition of the \"action\" to (h1 ^. h2) ^. g. But h1 ^. (h2 ^. g) = (h1 ^. h2) ^. g , by the asscociativity property of the group G. So condition (ii) is also satisfied and the defined rule is an \"action\" of H on G (proved).

(2) Here the new rule or \"action\" is defined by : h * g = g ^. h^-1 . The first condition is satisfied if g ^. e^-1 = g. This is however of course true, because by definition of identity element e^-1 = e, and since g belongs to the group G, g ^. e^-1 = g ^. e = g. Let us now check condition (ii).

LHS : h1 * (h2 * g) = h1 * (g ^. h2^-1) = g ^. h2^-1. h1^-1 (by the definition of the action)

RHS : (h1 ^. h2) * g = g ^. (h1 ^. h2)^-1 = g ^. h2^-1. h1^-1 ( by using the group inversion rule ---> (h1 ^. h2)^-1 = h2^-1. h1^-1 )

Hence LHS = RHS, so this rule also is an action of H on G. (proved)

(3) The group of integers Z, is a group under the usual group operation \"addition\" . The action adds an element from the subgroup 3Z to any integer belonging to Z. The result of summing to integers is always an integer. So it again belongs to Z. Let \'n\' be any element of Z. Then the orbit of \'n\' is denoted by n + 3k, where \'k\' is also an integer. There are infinite such orbits under this action of 3Z on Z.