Abstract finite group vs. finite group

Question

What is the difference between the definitions of abstract finite group and finite group?

I have some exposure to the finite group theory. But I came to know about abstract finite group from Constructive Approach to Automorphism Groups of Planar Graphs by Klavík et al. The authors used the idea while mentioning Frucht’s Theorem which, afaik, is applicable to all finite groups. So, I do not know why the authors used abstract finite group instead of finite group.

In A practical method for enumerating cosets of a finite abstract group Todd et al. define abstract finite group but the definition looks exactly same as the definition of finite group.

My question:

Should I assume that abstract finite group = finite group?

Answer 1

There is no definition of an "abstract finite group"; the term has no precise mathematical meaning. Generally speaking, if a mathematician refers to an "abstract group", they probably just mean "group", and are including the word "abstract" for emphasis, to clarify that any group is allowed, rather than some particular special kind of group that they have been talking about recently. This is how the term is used in the first paper. The second paper talks about "abstract definitions of groups", which are called presentations of groups in more modern language. The phrase "abstract group" itself only appears once in the entire paper (outside the title), and in context is referring to the group given by a particular abstract definition.

Answer 2

Yes. A little easier example may help. Consider the two finite groups:

$\mathbb Z/4\mathbb Z$ - the integers modulo 4

$\{1,-1,i,-i\}$ - the set of 4th roots of 1

These "look" very different, right? The first involves addition and remainders; meanwhile, the second has us multiplying complex numbers. However, they have the same "structure."

What do we mean by structure? In both sets, there is an way to compose (add/multiply) two elements and get a third; the each have identity element; both sets have one element of order one, one element of order two, and two elements of order 4. So, both sets as groups are essentially the same!!

Defining finite groups abstractly (read: axiomatically) helps to highlight the underlying structure. It takes away the irrelevant data.