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"