Elementary Properties of Groups 4 Basic Properties 1

Elementary Properties of Groups

4 Basic Properties 1. 2. 3. 4. • • • Uniqueness of the Identity Cancellation Uniqueness of Inverses Shoes and Socks Property What: Understand the property Why: Prove the property How: Use the property

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'.

To use uniqueness of identity • If ax = x for all x in some group G. • Then a most be the identity in G! *mod 40 5 15 25 35 5 15 15 35 25 15 5 25 5 15 25 35 35 15 5 35 25 Find e. e = 25!

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)

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.

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

When not to use cancellation • In D 4 • • R 90 D = D'R 90 You cannot cross cancel, since D ≠ D' Order matters!

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.

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