inverse element
An inverse element is the “undo” of another element: combine the two and you land exactly on the identity, as if nothing had happened. If a move rotates a square 90 degrees clockwise, its inverse rotates 90 degrees counter-clockwise. The whole point of an inverse is to cancel out and return to the starting point.
Formally, in a group with identity e, the inverse of an element a is an element b such that a * b = b * a = e. The fourth group axiom guarantees every element has one. With associativity in hand, the inverse is unique, so we may safely write it a⁻¹ (or -a when the operation is written additively).
Inverses are what make groups about reversible processes — every action can be undone. Be careful: the inverse depends on the operation. Under addition the inverse of 5 is -5; under multiplication the inverse of 5 is 1/5. And the number 0 has an additive inverse but no multiplicative inverse, which is exactly why you cannot divide by zero.
In the integers mod 7 under multiplication, the inverse of 3 is 5, because 3 · 5 = 15 = 14 + 1 ≡ 1 (mod 7). So 5 “undoes” multiplication by 3 in this group.
3⁻¹ = 5 in (ℤ/7ℤ)*.
A handy identity for products: (a * b)⁻¹ = b⁻¹ * a⁻¹, with the order reversed — like taking off your shoes and socks in the opposite order you put them on. The order matters in non-abelian groups, so do not write a⁻¹ * b⁻¹ by reflex.