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).
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Looking around, the narrator recognised specific items of crockery and linen that had once belonged to her mother, now set out alongside furniture that clearly did not match them. The arrangement was careless rather than deliberate, as though the pieces had simply been absorbed into daily …