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Math 201 Algebra I Notes
Definition 1. Algebra The study of structure that comes from endowing operations on the set.
Definition 2. Group A group is a pair (G, ฯ) where G is a set and ฯ is a binary operation ฯ : G ร G โ G which is associative and there is an element 1 โ G such that โx โ G, ฯ(1, x) = ฯ(x, 1) = x, and (โx โ G)(โy โ G)(ฯ(x, y) = ฯ(y, x) = 1).
Typically, we write a ยท b or ab for ฯ(a, b) and xโ^1 for the such that ฯ(x, y) = ฯ(y, x) = 1.
Example 1. (Z, +), (Q, +), (R, +), (C, +) are groups under addition. (Q{ 0 }, ยท), (R{ 0 }, ยท), (C
{ 0 }, ยท) are groups under multiplication.
Definition 3. Abelian Group A group is abelian if its operation is commutative.
Example 2. Let S be a set. Perm(S) := {bijectionsS โ S}. Perm(S) is a group under function composition.
Composition gives a well-defined (closed) operation Perm(S) ร Perm(S) โ Perm(S). If |S| < โ, then |Perm(S)| = (|S|)! If X is a set with |X| = n then write Sn for Perm(X) Sn is the symmetric group.
Example 3. S 3 write { 1 , 2 , 3 } for the underlying set. The following is an example of composing two elements in S 3. ( 1 2 3 1 3 2
Definition 4. Cayley Graph Have one vertex for each group element, arrows indicating what action each group element has on each other one.
Example 4. (Z, +) = G
Thus 1 connects the entire group.
Example 5. Cayley Group on S 3
Edge labeled * are under the operation of (23), while all the other edges are under the operation of (123).
Example 6. GLn(R) := group of invertible n ร n matrices with entries in R under multiplica- tion. GL means general linear group.
Example 7. Let T be a regular nโgon in R^2 centered at (0, 0). D 2 n := the group of rigid symmetries of T which fix T (as a set).
Definition 5. Dihedral Group The groups described in Example 7 are known as the Dihedral Groups.
Note: D 2 n can be viewed as a subgroup of GLn(R).
The edge 1 โ 2 can go to 1 โ 2 , 2 โ 3 , 3 โ 4 , 4 โ 5 , 5 โ 6 or you can flip it and it can goto 2 โ 1 , 3 โ 2 , 4 โ 3 , 5 โ 4 , 6 โ 5. It turns our that S 3 and D 6 are the same group. September 11, 2006
Definition 6. Let H โ G, then H is a subgroup of G if H is a group in its own right under the operation of G. This is denoted H โค G.
Theorem 1. In a group G, if H โ G, H 6 = โ
then H โค G โโ (โx, y โ H)(yโ^1 x โ H).
If partition {Sฮฑ} arises from an equivalence relation, then we will call each Sฮฑ an equivalence class.
Example 10. In Z, pick m โ Z and define
a โก b โโ m|a โ b
. Note that mZ โค Z so that a โก b โโ (โb) + a โ mZ Group G and subgroup H โค G define an equivalence relation by x โก y โโ yโ^1 x โ H. Actually, yโ^1 x โ H โโ x โ yH. So, x โก y โโ x โ yH. If H โค G, then a coset of H is a subset of G of the form yH for some y โ G.
Theorem 5. The cosets of H in G partition G.
