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
The proofs use the definition of groups.

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!
*mod 40 5 15 25 35

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