1. Groups have four basic properties: uniqueness of the identity, cancellation, uniqueness of inverses, and the "shoes and socks" property relating inverses of products. 2. Uniqueness of the identity means that a group has only one identity element. Cancellation means that if ba=ca then b=c, and similarly for left cancellation. 3. Uniqueness of inverses means that for each element a, there is a unique inverse b such that ab=ba=e. The "shoes and socks" property states that the inverse of a product ab is b^-1a^-1.

1. Elementary Properties of
Groups

2. 4 Basic Properties
1. Uniqueness of the Identity
2. Cancellation
3. Uniqueness of Inverses
4. 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!
*mod 40 5 15 25 35
5 25 35 5 15
15 35 25 15 5
25 5 15 25 35
35 15 5 35 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