Study Guide for Group Theory Midterm 1

This is my attempt to make a sexy study guide for my upcoming Group Theory Midterm. There'll be 3 parts: 1) Definitions, 2) Proofs, and 3) Properties of Groups Z(n), U(n), S(n), A(n), and D(n), with special focus on whether a given group is Abelian, cyclic, normal, etc.
This is the first section, Definitions.
I am expected to know the definitions of the following terms:
1: Even and odd permutations
2: Group isomorphism
3: Coset of H in G
4: External direct product
5: Normal subgroup.
So, here goes.
1: Even and odd permutations
A permutation is even if it is the product of an even number of 2-cycles and odd if it is the product of an odd number of 2-cycles. Specifically, if a permutation has an even number of elements, then it is an odd permutation, and if it has an odd number of elements, it is an even permutation.
2: Group isomorphism
An isomorphism P from a group G to a group G* is a one-to-one mapping from G onto G* that preserves the group operation. That is to say, an isomorphism is a function from one group to another and has three properties:
a. P is one-to-one; that is, every element g in G maps to a distinct element g* in G*.
b. P is onto; that is, every element g* G* is mapped to by a distinct element g in G*.
c. P is operation preserving (O.P.); that is, for all a,b in G, P(ab)=P(a)P(b).
Here are some notable properties of isomorphisms acting on elements and groups:
a. P maps the identity e in G to e in G*.
b. For every integer n and for every element a in G, P(a^n)=P(a)^n.
c. G is Abelian iff G* is Abelian.
d. G is cyclic iff G* is cyclic.
e. |a|=|P(a)|. Then, if G is finite, then G and G* have exactly the same number of elements of each order.
f. If K is a subgroup of G, then P(K) is a subgroup of G*.
3: Coset of H in G
Let G be a group and H be a subgroup of G. Then, for any a in G, the set {ah| h is in H} is denoted aH and is called the left coset of H in G containing a. A similar definition exists for the right coset.
Here are some properties of cosets:
a. a is in aH. Always, kids. Always.
b. aH=H iff a is in H (note that this makes aH a subgroup G, since H is a subgroup of G; this is the ONLY time that aH is a subgroup of G).
c. aH=bH iff a is in bH (or, similarly, iff (a^-1)b is in H).
d. EITHER aH=bH OR aH&bH={0}. Super important to know this. Super crucial.
e. aH=Ha iff H=aH(a^-1).
4: External direct product
Let G1, G2,..., Gn be a finite collection of groups. The external direct product of
G1, G2,..., Gn is denoted for this blog as G1#G2#...#Gn is the set of all n-tuples for which the ith component is an element of Gi and the operation is componentwise.
What does all that mean? It means that the external direct product is the set of n-tuples (like an ordered pair, but an ordered... n-tuple) where the first element is produced by using the operation of G1 on all of the "first" elements, the second element is produced by using the operation of G2 on all of the "second" elements, etc. This is weird. I don't quite get it. I'll return to this.
A few things to know about external direct products:
a. The order of an element in a direct product of a finate number of finite groups is the lcm of the orders of the components of the elements.
b. If G, H are cyclic, then G#H is cyclic iff |G| and |H| are relatively prime.
5. Normal subgroups
A subgroup H of a group G is called a normal subgroup of G if aH=Ha for all a in G. This is denoted by H|>G, for this blog.
The normal subgroup test:
A subgroup H of G is normal in G iff xH(x^-1) <= H for all x in G.
Ok, I'm tired, so I'm going to read myself to sleep now. A couple of proofs tomorrow, probably. Probably Disjoint Cycles Commute and the Four Corollaries to Lagrange's Theorem. Fun stuff.