Elementary Properties of Groups

1. Elementary Properties of Groups

2. 4 Basic Properties • Uniqueness of the Identity • Cancellation • Uniqueness of Inverses • Shoes and Socks Property • What: Understand the property • Why: Prove the property • How: Use the property

3. 1. Uniqueness of Identity • In a group G, there is only one identity element. • Proof: Suppose both e and e' are identities of G. Then, 1. ae = a for all a in G, and 2. e'a = a for all a in G. Let a = e' in (1) and a = e in (2). Then (1) and (2) become (1) e'e = e', and (2) e'e = e. It follows that e = e'.

4. To use uniqueness of identity • If ax = x for all x in some group G. • Then a most be the identity in G! Find e. e = 25!

5. 2. Cancellation • In a group G, the right and left cancellation laws hold. That is, ba = ca implies b = c (right cancellation) ab = ac implies b = c (left cancellation)

6. Proof: Right cancellation • Let G be a group with identity element e. Suppose ba=ca. Let a' be an inverse of a. Then (ba)a' = (ca)a' => b(aa') = c(aa') by associativity => be = ce by the definition of inverses => b = c by the definition of the identity.

7. Proof of left cancellation • Similar. • Put it in your proof notebook.

8. When not to use cancellation • In D4 R90D = D'R90 • You cannot cross cancel, since D ≠ D' • Order matters!

9. 3. Uniqueness of inverses • For each element a in a group G, there is a unique element b in G such that ab=ba=e. • Proof: Suppose b and c are both inverses of a. Then ab = e and ac = e so ab = ac. Cancel on the left to get b = c.

10. 4. Shoes and Socks • For group elements a and b, (ab)-1 =b-1a-1 • Proof: (ab)(b-1a-1) = (a(bb-1))a-1 =(ae)a-1 = aa-1 = e Since inverses are unique, b-1a-1 must be (ab)-1