Which group axiom says that \(a*b\in G\) whenever \(a,b\in G\)?

Write the associativity axiom for a binary operation \(*\).

For multiplication on \(\{1,-1\}\), what is the identity element?

For multiplication on \(\{1,-1\}\), what is the inverse of \(-1\)?

Show that \(\{1,-1\}\) is closed under multiplication.

For \(G=\{0,1\}\) under addition modulo \(2\), list the inverse of each element.

Why does one successful example of closure not prove the closure axiom for an infinite set?

A set has a left identity \(e*a=a\) for every \(a\). Is that alone enough for the identity axiom?

Test associativity for subtraction using \(a=8\), \(b=3\), and \(c=2\).

Show that the odd integers are not closed under addition.

For the integers under addition, verify the inverse axiom for an arbitrary integer \(a\).

Explain why associativity for \(\{1,-1\}\) under multiplication can be inherited from real multiplication.

Check all four group axioms for \(\{1,-1,i,-i\}\) under complex multiplication.

The set \(\{0,1,2,3\}\) uses addition modulo \(4\). Verify that every element has an inverse.

A binary operation on \(G\) has a table whose entries all lie in \(G\). Which axiom is proved, and which axioms remain?

Suppose \(e\) is an identity and \(a*b=e\). Why is this not always enough to prove \(b\) is the inverse of \(a\)?

For \(G=\mathbb R\) with \(a*b=a+b+ab\), show why \(-1\) causes the inverse axiom to fail on all real numbers.

Give a proof that closure and inverses imply the identity lies in a non-empty subgroup candidate \(H\) of a group \(G\).

A student checks closure, identity, and inverses for a new operation, then says associativity is obvious because the operation is commutative. Diagnose the error.

Why is finding a different identity-like element for each \(a\in G\) not enough for the identity axiom?