Why does Dummit and Foote specify the identity of a group $G$ as $1$?

Question

A group need not necessarily be composed of integers.

Why does Dummit and Foote specify the identity as $1$?

Consider $G = \{\text{Jim}, \text{Lara}, \text{Kan}\}$ s.t. the identity of $G$ is $\text{Lara}$.

Answer 1

For the same reason we write the group operation $\cdot$ or $+$, it is because we are abstracting the ideas of our normal number systems into groups/rings/fields. It makes sense because we are redefining "$1$" to be simply the multiplicative identity, and then the intuition of how multiplicative identites work in $\mathbb R$ or $\mathbb Q$ actually gives good intuition for the general group setting.

I will add two things: not all concepts carry over, for example (even though Lang seems to do it) thinking about a group operation as $+$ when the operation isn't abelian usually breaks intuition, so we use $\cdot$ generally (because we can easily think of a classical non-communative multiplication: matrix multiplication). Secondly, for (finitely generated) abelian groups, it actually turns out that every group looks exactly like a bunch of copies of $\mathbb Z$ (called the free part) together with copies of the integers modulo some set of numbers (called the torsion part).

Answer 2

Just to give a different viewpoint to Juan.

We have not developed enough symbols to give every distinct concept its own symbol. Some reuse is necessary. In general, you need to watch the context to know the meaning of the symbol. Even within a particular context, you need to check what the author means by it.

You might be more familiar with the identity of a group being denoted by $e$ but this is also commonly used for the base of natural logarithms and frequently just for an arbitrary number. If you are reading a paper on group theory then $e$ is probably the group identity. If the paper is on calculus then $e$ is more likely to be the base of natural logarithms.

As mentioned in comments, $1$ in the Natural numbers, the Integers, the Rational numbers, the Real numbers, and the Complex numbers is not technically the same thing but few people get confused or upset by that.

Using $1$ for the identity of a group in which the operation is denoted as multiplication is attractive. I am surprised that is not more common.

$1$ is typically used for the multiplicative identity of a field or a ring. Also $2$ is commonly used for $1 + 1$, $3$ for $2 + 1$ etc. More care is needed with these since, in an arbitrary ring or field, they are not all necessarily non-zero.

$\pi$ is probably the most consistently used symbol but even it is sometimes used for things very different from the famous number.