Elementary Properties of Groups

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 The proofs use the definition of groups.

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! Find e. e = 25! *mod 40 5 15 25 35

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 D4 R90D = D'R90 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-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