You have a set. Maybe it's the integers, maybe it's the points on a circle, maybe it's the symmetries of a square. You also have a rule for combining two elements to get a third — call it multiplication, addition, or composition. That structure is a group.
Now here is the question that leads to character identification: If I know how the group behaves, what are all the ways I can map it into the complex numbers (the number line with −1) while respecting the group operation?
Why complex numbers? Because they rotate. Multiplying by eiθ is a rotation. And rotations are the simplest non-trivial symmetries. So a character is a homomorphism from a group G to the multiplicative group of non-zero complex numbers, C×.
Note
A homomorphism means: if you multiply two group elements first and then map, you get the same result as mapping each element first and then multiplying the images. In symbols: χ(gh)=χ(g)χ(h) for all g,h∈G.
Intuition: What does a character "see"?
Imagine you have a finite group G — say the cyclic group of order n, written Zn. You can think of its elements as rotations of a regular n-gon by multiples of 360∘/n. A character assigns to each rotation a complex number of magnitude 1 (a point on the unit circle) such that composing two rotations corresponds to multiplying their assigned numbers.
For Zn, the characters are exactly the maps
χk(m)=e2πikm/n,k=0,1,…,n−1.
Why? Because if you rotate by m steps and then by m′ steps, you've rotated by m+m′ steps. The character must satisfy
χ(m+m′)=χ(m)χ(m′).
The only way to do that with complex numbers of magnitude 1 is with an exponential. The parameter k tells you how many times the character "winds around" the circle as you go through the group.
Tip
For a cyclic group, characters are just the n-th roots of unity, each raised to the power of the group element. The k=0 character sends everything to 1 — that's the trivial character.
Precise statement
Let G be a finite group. A character of G is a group homomorphism
χ:G→C×.
The set of all characters of G is denoted G and is itself a group under pointwise multiplication:
(χψ)(g)=χ(g)ψ(g).
This group G is called the dual group of G.
For a finite abelian group, G is isomorphic to G itself — but not in a canonical way. For a non-abelian group, characters are more subtle: you need to consider traces of matrix representations, not just numbers. But the core idea remains: characters are the simplest ways to "hear" the group structure through complex numbers.
Important
Characters are multiplicative maps: χ(gh)=χ(g)χ(h). They always send the identity to 1, and they send inverses to complex conjugates (since ∣χ(g)∣=1 for finite groups).
The address had drawn the narrator back to reclaim her mother's belongings, but her visit taught her that those objects could no longer mean anything to her. …
Because the belongings had lost their meaning severed from her old life, brought only pain, and clinging to them would keep her chained to a lost past.
When the narrator finally visited Number 46, she found all her mother's possessions crammed into Mrs Dorling's living room in an ugly, tasteless arrangement, filling the air with a “muggy smell.” Instead of comfort, the sight “oppressed” her; she stood “in a room I knew and did not know,” and even her own voice sounded strange. She realised a hard truth: “the objects that are linked in your memory with the familiar life of former times instantly lose their value when, severed from them, you see them again in strange surroundings.” …