Basic Properties of Groups

From the definition of a group, the following can be proven in general.

Uniqueness of Identity

If e and e′ are both identities for a group G, then:

e=e′.

Consider a group G with two identities e and e′. Then, from the definition of the identity:

e∗e′=e and e∗e′=e′⟹e=e′

so the identity is unique.

Uniqueness of Inverse

Given a∈G, there exists a unique element denoted by a−1 which satisfies

a∗a−1=a−1∗a=e

for the identity e.

Let b and b′ be inverses of a in a group. This implies that a∗b=e and a∗b′=e for the identity e.

Therefore:

a∗b=a∗b′(b∗a)∗b=(b∗a)∗b′e∗b=e∗b′b=b
(a−1)−1=a

For any a∈G, we have that:

(a−1)−1=a.

Since a−1 has an inverse of (a−1)−1, it follows that (a−1)−1∗a−1=e. Then by multiplying on the right by a we find that:

(a−1)−1∗(a−1∗a)=e∗a(a−1)−1∗e=a(a−1)−1=a
(a∗b)−1=(b−1∗a−1)

For any a,b∈G, we have that:

(a∗b)−1=(b−1∗a−1).
(b−1∗a−1)∗(a∗b)=b−1∗(a−1∗a)∗b=b−1∗e∗b=b−1∗b=e
Cancellation

Given a,b,c∈G:

a∗b=a∗c⟹b=c.

Cancellation from both sides follows very easily from the existence of an inverse. Here is just the case on the left hand side:

a∗b=a∗c(a−1∗a)∗b=(a−1∗a)∗ce∗b=e∗cb=c