As promised, we're now going to learn about subgroups. Subgroups are extremely important in group theory even though the definition is quite simple.
Definition: Subgroup
If G is a group and H is a subset of G, then H is a subgroup of G if H is a group itself under the operation of G.
If G is a group and H is a subset of G, then H is a subgroup of G if H is a group itself under the operation of G.
If H is a subgroup of G, all that means is that H is a group under the operation of G and H lives inside of G - that is every element of H is also an element of G. Like I said before, this is a very simple definition, but the subgroups are often one of the most interesting parts of a group. Lets look at an example of a subgroup.
Example: 2Z is a subgroup of Z
Define Z to be the set of integers. We saw earlier that Z is a group under standard addition. Now lets define 2Z to be the set of all even integers - that is 2Z = {...,-6,-4,-2,0,2,4,6,...}. First, notice that H is a subset of G (we denote this by H⊆G). I now wish to convince you that H is a subgroup of G by showing that H is also a group under standard addition. The properties of identity and associativity are inherited from Z - that is that ∀a,b,c∈2Z, (a+b)+c = a+(b+c) because a,b,c∈Z and similarly, ∀a∈2Z, a+0 = a because a∈Z. Inverses come pretty easily in this case, too, because ∀a∈2Z, -a∈2Z and a+(-a) = 0. Finally, closure comes from the fact that the sum of two odd numbers must be odd, so a,b∈2Z automatically implies that a+b∈2Z. Thus 2Z is a group under standard addition and 2Z is a subgroup of Z.
Define Z to be the set of integers. We saw earlier that Z is a group under standard addition. Now lets define 2Z to be the set of all even integers - that is 2Z = {...,-6,-4,-2,0,2,4,6,...}. First, notice that H is a subset of G (we denote this by H⊆G). I now wish to convince you that H is a subgroup of G by showing that H is also a group under standard addition. The properties of identity and associativity are inherited from Z - that is that ∀a,b,c∈2Z, (a+b)+c = a+(b+c) because a,b,c∈Z and similarly, ∀a∈2Z, a+0 = a because a∈Z. Inverses come pretty easily in this case, too, because ∀a∈2Z, -a∈2Z and a+(-a) = 0. Finally, closure comes from the fact that the sum of two odd numbers must be odd, so a,b∈2Z automatically implies that a+b∈2Z. Thus 2Z is a group under standard addition and 2Z is a subgroup of Z.
Notice here that 2Z is itself a group under standard addition. It is a subgroup relative to Z because 2Z⊆Z and because 2Z is a group under the same operation of Z.
As you read the example and my long-winded justification for why 2Z is a subgroup of Z, there are a couple of things that might pop out. First, associativity and identity were automatically inherited and didn't require any justification. Also, the explanation was really awkward and long and cumbersome. There is a much better way to prove when something is a subgroup and that is what I'm going to go over in my next post.
References
Previous Related Post: Early Properties of Groups
Text Reference: Gallain Chapter 3
Wikipedia: Subgroup
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