Algebraic structures: monoids, groups and rings

Which of these are groups? (Choose all that apply.)

The operation $a \ast b = \gcd(a, b)$ on $\N = \{0, 1, 2, \dots\}$ is associative and commutative. Which natural number $e$ is its identity element, i.e. satisfies $\gcd(e, a) = a$ for every $a \in \N$?

Find the inverse of $7$ in $\Z/26\Z$: the number $x \in \{1, \dots, 25\}$ with $7x \equiv 1 \pmod{26}$. (This is the key step in decoding an affine cipher.)

The proof that inverses in a monoid are unique reads: if $ba = 1$ and $ac = 1$ then $b = b1 = b(ac) = (ba)c = 1c = c$. Which equality uses associativity?

Between groups, any $f$ with $f(ab) = f(a)f(b)$ automatically satisfies $f(1) = 1$. Which example shows that this fails for monoids, so that $f(1) = 1$ must be required separately?

$\Z/91\Z$ is not a field. Find the smallest $x \in \{1, \dots, 90\}$ with $7x \equiv 0 \pmod{91}$, which exhibits $7$ as a zero divisor.

Which Lean statement correctly formalises "in a monoid, a left inverse and a right inverse of the same element are equal"?

Write is_group(elements, op) that returns True exactly when op (a Python function of two arguments) is a group operation on the finite list elements: check closure, associativity, an identity, and inverses. Then test it on addition modulo 5, multiplication modulo 6, and multiplication modulo 12 on the units $\{1, 5, 7, 11\}$.