Group axioms and examples Quiz

1. Which axiom fails for $\{1,2,3,4,5\}$ under multiplication modulo 6?

• A. Associativity
• B. Identity
• C. Closure
• D. None: it is a group

2. What is the inverse of $4$ in $\{0,1,\ldots,6\}$ under addition modulo 7?

• A. $4$
• B. $2$
• C. $3$
• D. $5$

3. A Cayley table has an element repeated in one row. What does this show?

• A. The group is abelian
• B. The set is not a group
• C. The group is cyclic
• D. The identity is missing but the set is still a group

4. Which element generates $\{0,1,2,3,4,5,6,7\}$ under addition modulo 8?

• A. $2$
• B. $4$
• C. $6$
• D. $3$

5. Which set of matrices forms a group under matrix multiplication?

• A. All $2\times2$ matrices
• B. All non-singular $2\times2$ matrices
• C. All $2\times2$ matrices with determinant $0$
• D. All $2\times2$ matrices with a zero entry