Abstract Algebra: Using subsets to handle subgroups in a good way

In Abstract Algebra we usually care only that a subset of the domain/carrier of a Group forms a subgroup of that group and not so much for the actual subgroup.

In this post, I’ll use the following definition for a Group.

Let G be a set. Let e be an element of G. Let op be a binary operation over G. Let inv be a unary operation over G. We say that <G, e, op, inv> is a Group if:

(∀ a ∊ G)(∀ b ∊ G)(∀ c ∊ G)[op(a, op(b, c)) = op(op(a, b), c)];

(∀ a ∊ G)[op(e, a) = a & op(a, e) = a];

(∀ a ∊ G)[op(a, inv(a))= e & op(inv(a), a) = e].

You can find my reasons why in my previous blog post.

So here is what I’m gonna do: instead of defining what a subgroup is I’ll say what is it for a subset of the domain/carrier of a Group to form a subgroup of that Group.

But first, let's say what the domain/carrier of a Group really is.

If 𝓖 = <G, e, op, inv> and 𝓖 is a Group then G is the domain/carrier/Universum of the Group 𝓖. The domain/carrier/Universum is actually the set of the elements… Also if 𝓖 is a Group we will denote the Universum of 𝓖 with U(𝓖 ).

We move on to say what is it for a subset of the Universum of a Group to form a subgroup of that Group.

Let 𝓖 = <G, e, op, inv> and let 𝓖 be a Group. Let H be a subset of G. We say that H forms a subgroup of 𝓖 if <H, e, op↾ H x H, inv ↾ H> is a Group.

In the above definition op↾ H x H and inv↾ H are the restrictions of op and inv only to the elements of H.

Let's examine what it really means for a subset of the Universum of a Group to form a subgroup of that Group.

Let 𝓖 = <G, e, op, inv> and let 𝓖 be a Group. Let H be a subset of G. In order <H, e, op↾ H x H, inv ↾ H> to be a Group it must be true that:

1. e is an element of H;
2. op↾ H x H is a function from H x H to H;
3. inv ↾ H is a function from H to H;
4. op↾ H x H is asociative;
5. e is the neutral (identity) element of H with respect to op↾ H x H;
6. inv ↾ H gives the inverse element to each element of H with respect to op↾ H x H.

Well the aboves are true iff it is true that:

1. e ∊ H;
2. Range(op↾ H x H) ⊆ H;
3. Range(inv ↾ H) ⊆ H.

Now if 𝓖 is Group and H forms a subgroup of 𝓖 we can easily denote that with H ⪗ 𝓖 for an example.

Now you may ask what did we achieve with all this?

The answer is very simple: We can now properly work with subsets and at the same time retain the connection with the Group of which the subset forms a subgroup.