Proof techniques in abstract algebra

Basic techniques

We wish to prove an element is unique

We have standard method that assuming there are two elements and we show that they are the same using some assumptions or premises.

How to Show Two Binary Structures Are Isomorphic

Let $(S, *)$ and $(S’, *’)$ be binary algebraic structures.
To prove that $(S, *) \cong (S’, *’)$, proceed as follows.

Define the map.
Define a function \[ \phi : S \to S’ \] by explicitly specifying $\phi(s)$ for every $s \in S$.
Injectivity (one-to-one).
Show that $\phi$ is injective.
That is, assume \[ \phi(x) = \phi(y) \] for $x,y \in S$, and deduce that $x = y$.
Surjectivity (onto).
Show that $\phi$ is surjective.
That is, let $s’ \in S’$ be arbitrary and prove that there exists $s \in S$ such that \[ \phi(s) = s’. \]
Homomorphism property.
Show that $\phi$ preserves the operation: \[ \phi(x * y) = \phi(x) *’ \phi(y) \quad \text{for all } x,y \in S. \] This is typically verified by direct computation.

If all four steps are satisfied, then $\phi$ is an isomorphism and the structures $(S, *)$ and $(S’, *’)$ are isomorphic.

Structural Properties and Isomorphism

A structural property of a binary structure $(S, )$ is a property that is *preserved under isomorphism. That is, if \[ (S, *) \cong (S’, *’), \] then $(S, *)$ and $(S’, *’)$ must share all structural properties.

Structural properties are not concerned with:

the names of the elements of the set,
the symbols used to denote the operation,
whether one set is a subset of another.

Instead, structural properties depend only on the algebraic structure itself.

Examples of structural properties:

Cardinality (finite, countably infinite, uncountable)
Existence of an identity element
Existence of inverses
Associativity and commutativity
Number of elements in the underlying set

Non-structural properties:

Element labels or notation
The name of the operation (e.g., $+$ vs. $\times$)
Whether one structure is a proper subset of another

Key consequence:
If two structures have different cardinalities, they cannot be isomorphic.

For example, \[ (\mathbb{Q}, +) \not\cong (\mathbb{R}, +), \] since $\mathbb{Q}$ is countable while $\mathbb{R}$ is uncountable.

Important caution:
A structure may be isomorphic to a proper subset of itself.
Therefore, being a proper subset does not rule out isomorphism.

Summary:
Isomorphism preserves structure, not appearance.