# Who in history coined the term “character” of a group and why is it called so?

I first read the term in an introduction of Fourier transform on locally compact groups. In this article on Character of a group from Encyclopedia of Mathematics, a character of a group is defined as a homomorphism of the given group into some standard Abelian group $$A$$. It is said that

The concept of a character of a group was originally introduced for finite groups $$G$$ with $$A=T:=\{z\in\mathbb{C}:|z|=1\}$$ (in this case every character $$G\to \mathbb{C}^*$$ takes values in $$T$$).

Who in history coined the term "character" of a group and why is it called so?

• Check Vorlesungen über Zahlentheorie on Google books By Peter Gustav Lejeune Dirichlet (1894). He introduced the term character and class which is now used group theory. Search the word "Charaktere" pg 612. I used Deepl.com to translate. – M. Farooq Apr 26 at 23:47
• @M.Farooq：Thanks for that. There is also an English edition: books.google.ca/books/about/… – Mars Apr 30 at 11:28

With credit to @ConsigliereZARF for helpful comments and references. The earliest definition of group character ("Charakter") for Abelian groups is likely due to Weber (1881-2), and it was generalized to general groups by Frobenius (1896). According to Mackey's survey Harmonic analysis as the exploitation of symmetry:

"In 1881 Weber defined a character of a finite commutative group $$G$$ to be a complex valued function $$\chi$$ on $$G$$ such that $$\chi(xy) = \chi(x)\chi(y)$$ for all $$x$$ and $$y$$ in $$G$$. This definition was an abstract generalization of one given three years earlier by Dedekind in connection with his work on algebraic number theory, which was inspired in turn by early work of Gauss and Dirichlet (see sections 6 and 12). While Weber's definition makes sense for arbitrary finite groups, it is more or less vacuous except insofar as the group has commutative aspects. Specifically, every character is identically one on the commutator subgroup and consequently the only characters not identically one are derived trivially from characters of commutative quotient groups.

Group theory acquired a powerful new tool that was soon to become almost indispensable when G. Frobenius (1849-1917) published a paper in 1896 showing that there is a natural generalization of the character notion that involves the whole group $$G$$ in a significant and interesting way—even when $$G$$ is non-commutative."

Mackey appears to refer to Weber's Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist (1882), where he generalizes Dirichlet's prime number theorem. Weber introduces the character using a basis of Abelian group and remarks that it always satisfies the multiplicative property. In a footnote he credits Dedekind's Supplement XI to the third edition of Dirichlet's Vorlesungen über Zahlentheorie (1879), p. 581.

Dirichlet uses the word "Charaktere" throughout Vorlesungen über Zahlentheorie, the earliest on p.77 in connection with quadratic reciprocity, and then expands the use of the name (p. 316). Dirichlet's characters are not quite group characters, and the notion of a group was not in place at the time of Gauss's or Dirichlet's writing. However, concerning Gauss's "total character" of a quadratic form from Disquisitiones Arithmeticae (1801), Mackey writes:

"The word character as used today stems directly from Gauss's use of the term in his theory of binary quadratic forms... Dedekind makes the same definition for the special case of the ideal class group of an algebraic number field. As Dedekind himself had pointed out earlier, the ideal class group is a generalization of Gauss's group of equivalence classes of binary quadratic forms. Dedekind's mode of expression is such as to make it clear that he regards his definition as a generalization of that of Gauss."

Before Gauss, quadratic characters, later generalized by Jacobi and Dirichlet, appear in Legendre's Essai sur la theorie des nombres (1798). Introducing the quadratic character symbol, he simply wrote:

"As the quantities analogous to $$N^{\frac{c-1}2}$$ will frequently appear in the course of our research, we will use the abbreviated character $$\left(\frac{N}{c}\right)$$ to express the residue from dividing $$N^{\frac{c-1}2}$$ by $$c$$; the residue which, according to what we have just seen, can only be $$+1$$ or $$-1$$."[ See Cajori, History of Mathematical Notation, p.30]

So to him "character" meant nothing more than "symbol". Whether this is where Gauss took the word from is unclear, Mackey says that large parts of Disquisitiones Arithmeticae were written before Legendre's Essai was published, but Gauss "acknowledges that he was inspired to study quadratic forms by the work of Lagrange and Legendre". Legendre divided classes of forms into cosets that Gauss called genera, and forms were said to be in the same genus if they had the same "total character", a system of $$+1$$s and $$-1$$s canonically associated to them. Each $$\pm1$$ was produced by a function on forms what we today call a group character.

• No. The answer is well known to be Gauß (1801). – Consigliere ZARF Apr 29 at 14:41
• @Mars You just accepted a wrong answer. See Mackey 1980, pp. 555–556, 598. – Consigliere ZARF Apr 30 at 13:13
• @ConsigliereZARF: thanks for the link. The most relevant text I found is "It turns out that Gauss's characters are characters in the modern sense for the finite commutative group in question.", "The word character as used today stems directly from Gauss's use of the term in his theory of binary quadratic forms." and "Characters and Fourier analysis on finite commutative groups occur implicitly in other parts of Gauss's work. " – Mars Apr 30 at 20:48
• @ConsigliereZARF: it seems that the term had evolved a bit since Gauss. – Mars Apr 30 at 20:50
• I think Mackey refers to Weber's Beweis des Satzes, p. 307-8, where he indeed introduces characters in group theoretic terminology with reference to Dedekind's Supplement XI to the third edition of Dirichlet's Vorlesungen über Zahlentheorie (1879). This is earlier than Miller's references, and might be the earliest occurrence of group character proper. I will change the answer accordingly. – Conifold May 1 at 22:40