They're just spooky names for simple concepts - and the article defines them on first use. If abstract algebra were a requirement, they'd skip these definitions.

Paraphrasing 'Group' from the article to see if I've understood it:

A set of elements G, and some operation ⊕, where

  (g1 ⊕ g2) is also in G. // "Type-safety"

  Some g0 exists such that (gn ⊕ g0) == (g0 ⊕ gn) == gn // "Zero"

  For every g, there's some inverse gi such that (g ⊕ gi) == (gi ⊕ g) == g0 // "Cancelling-out"

  a ⊕ (b ⊕ c) == (a ⊕ b) ⊕ c // "Associative"

  If (a ⊕ b) == (b ⊕ a) then the group is also "abelian/commutative"

Is the aspirin symbol you're using as + figure, a special kind of +, or just a different looking +? What does the circle around the + mean?

I'm mentioning this, as other people in this thread are discussing "explaining symbols you use", and you're using a non-standard symbol for +. I can easily imagine a circle around + making + a different operation, and wonder if it is so?

Aspirin I've bought in the past has a + on it, and its trademark is a + within a circle. That's why I've latched on what a "common person" might view the symbol as:

https://www.brand.aspirin.com/sites/g/files/vrxlpx46831/file...

Interestingly, I have University level math courses, but decades out of date, and have never run into that symbol. I see it here:

https://en.wikipedia.org/wiki/Direct_sum

They're spooky names for simple concepts, with extremely deep consequences and hard theory, don't be fooled.