9.1 Subgroups
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The group (2\mathbb{Z},+) is a subgroup of (\mathbb{Z},+).
If n is a positive integer, then the group (n\mathbb{Z},+) is a subgroup of (\mathbb{Z},+).Here’s a simple description of what needs to be checked for a subset to be a subgroup.
Let G be a multiplicatively written group and let H\subseteq G be a subset of G, Then H is a subgroup of G if and only if the following conditions are satisfied.
(Closure) If x,y\in H then xy\in H.
(Identity) e\in H.
(Inverses) If x\in H then x^{-1}\in H.
We don’t need to check associativity, since x(yz)=(xy)z is true for all elements of G, so is certainly true for all elements of H. So the conditions imply H is a group with the same operation as G.
If H is a group, then (Closure) must hold. By uniqueness of identity and inverses, the identity of H must be the same as that of G, and the inverse of x\in H is the same in H as in G, so (Identity) and (Inverses) must hold.□
If H,K are two subgroups of a group G, then H\cap K is also a subgroup of G.
We check the three properties in the theorem.
If x,y\in H\cap K then xy\in H by closure of H, and xy\in K by closure of K, and so xy\in H\cap K.
e\in H and e\in K, so e\in H\cap K.
If x\in H\cap K, then x^{-1}\in H since H has inverses, and x^{-1}\in K since K has inverses. So x^{-1}\in H\cap K.
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