Groups, Rings and Fields are often referred to by the set involved without mention of the operation(s). For example, the "group (G,+)" may be called the "group G".

When did this practice originate and why?

It seems the labour saved in writing "G" instead of "(G,+)" (for example) is trivial compared with the loss of clarity and precision so I wonder what other reasons there could be?

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    $\begingroup$ One might first ask whether $G$ or $(G,+)$ came first. (I don't know, but I suspect the notation $(G,+)$ came later.) $\endgroup$
    – Michael E2
    Commented May 18, 2016 at 13:40
  • 3
    $\begingroup$ About "loss of clarity"... have you actually spent much time working with groups, rings, or fields? There is essentially never a problem caused by referring to a group, ring, or field by the name of the underlying set. In the rare situations when an actual confusion might arise (e.g., regarding the real numbers as a topological group with the discrete topology instead of its usual topology) a special one-time notation is used (e.g., $\mathbf R_d$ for that topological group). You are asking about a problem that is never a problem to those who work with groups, rings, and fields. $\endgroup$
    – KCd
    Commented May 21, 2016 at 3:09

1 Answer 1


The shift seems quite recent.

For the "short" writing, see some "old" Algebra's textbook, like:

Definition 1. A set with an associative law of composition, possessing an identity element and under which every element is invertible, is called a group. [...] If $G$ and $H$ are two groups, ...

Definition. A group $G$ is a system of elements with a binary operation which ...

From the '90s, we can see the "new wave" emerging:

  • Saunders Mac Lane & Garrett Birkhoff, Algebra (3rd ed 1991), page 43:

A group $G$ is a set $G$ together with a binary operation...,

but see page 85:

A ring $R=(K,+,∙,1)$ is a set $R$ with two binary operations...

and :

A group is an ordered pair $(G,∙)$ where $G$ is a set ...

  • 1
    $\begingroup$ I don't think this is addressing the question that was asked. The OP wants to know who first started using just the letter for the underlying set as the label for a group, ring, or field. The answer is that this has been done since these concepts were first defined. $\endgroup$
    – KCd
    Commented May 21, 2016 at 3:11

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