What data must be specified before the phrase "\((G,*)\) is a group" has a precise meaning?

State the four axioms in the definition of a group \((G,*)\).

In \((\mathbb Z,+)\), what is the identity element?

In \((\mathbb Z,+)\), what is the inverse of \(7\)?

Check closure for the set of even integers under addition.

Why is \((\mathbb N,+)\), with \(\mathbb N=\{1,2,3,\ldots\}\), not a group?

Is commutativity required in the definition of a group? Explain with the correct axiom it is often confused with.

The operation \(*\) on real numbers is defined by \(a*b=a+b+1\). Find the identity element if one exists.

For real numbers under \(a*b=a+b+1\), find the inverse of an element \(a\).

Explain why the non-zero real numbers form a group under multiplication.

Why do the real numbers fail to form a group under multiplication?

A set \(G\) with operation \(*\) has closure, an identity, and inverses. Why is this not enough to conclude it is a group?

Let \(G=\{0,1,2\}\) with addition modulo \(3\). Verify the group axioms concisely.

The positive real numbers are given the operation \(a*b=ab\). Verify that this is a group.

For \((\mathbb Z,-)\), identify which group axiom fails first in a standard check and explain.

On real numbers define \(a*b=a+b-ab\). Test whether \(1\) can belong to a group with this operation.

A student claims \((\mathbb Z,\times)\) is a group because multiplication is associative and has identity \(1\). Give the missing axiom check that disproves the claim.

Prove that a group has only one identity element.

Prove that an element of a group has only one inverse.

A finite operation table has entries only from \(G\), has a two-sided identity, and each element appears with the identity somewhere in its row and column. Why must associativity still be tested separately?