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Handout Wednesday 10/8
Last time we introduced the concept of a subgroup of a group. This is defined as a subset that is also a group under the same operation. We decided that to check whether a subset was a group, we need to check three properties: (1) \(e \in H\) (\(H\) contains the identity of \(G\)), (2) \(\forall a, b, \in H\) we have \(ab \in H\) (\(H\) is closed under the operation), and (3) \(\forall a \in H\) we have \(a\inv \in H\) (\(H\) is closed under inverses).
Note though that we still need the operation to be the same. In particular, \(\Z_4\) is not a subgroup of \(\Z_8\text{.}\)
A few examples of subgroups:
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Let \(G = \Z\text{,}\) the group of integers under addition. What are the subgroups? Is \(3\Z\) a subgroup? These are all the multiples of 3. Check the 3 things.
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Let \(G = \mathscr{F}(\R)\text{,}\) the group of all real-valued functions under addition. One subgroup is the set of all continuous functions. Also the set of all differentiable functions, or linear functions, or polynomials.
We would also like to say some things in general. For example, let’s prove that if \(G\) is any abelian group, then \(H = \{g^2 \st g \in G\}\) is a subgroup of \(G\text{.}\)
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Another way to write the subgroup: \(H = \{g \in G \st g=a^2 \text{ for some } a \in G\}\text{.}\)
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For closure: assume \(a, b \in H\text{.}\) That is, \(a = x^2\) and \(b = y^2\) for \(x, y \in G\text{.}\) What is \(ab\text{?}\) Well, \(ab = x^2y^2 = (xy)^2\) because \(G\) is abelian. But \(xy \in G\text{,}\) so \(ab \in H\text{.}\)
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For inverses: assume \(a \in H\text{.}\) This means, \(a = x^2\) for some \(x \in G\text{.}\) What about \(a\inv\text{?}\) Well, \(a\inv = (x^2)\inv = (x\inv)^2\text{,}\) and since \(x\inv \in G\) we see that \(a \in H\text{.}\)
By the way, what is \(H\) for \(\Z\) here?
Try two more: Finish the proof that \(Z(G) = \{c \in G \st cx = xc \text{ for every } x \in G\}\) is a subgroup of \(G\text{.}\) This is called the center of \(G\) (the set of elements that commute with everything).
Then prove that the centralizer of \(H\) in \(G\) is a subgroup: \(C(H) = \{g \in G \st ghg\inv = h \text{ for all } h \in H\}\text{.}\)
If there is time, consider \(\langle a \rangle = \{a^n \st n \in \Z\}\text{.}\) That is, the set containing all the positive and negative powers of \(a\text{.}\)
