Groups Quiz

1. Why is $(\mathbb{Z},\times)$ not a group?

• A. It is not closed.
• B. It has no identity.
• C. It is not associative.
• D. Most elements have no inverse in $\mathbb{Z}$.

2. What is the order of the group of symmetries of a square?

• A. $4$
• B. $6$
• C. $8$
• D. $16$

3. How many proper non-trivial subgroups does $\{1,5,7,11\}$ under multiplication modulo $12$ have?

• A. $3$
• B. $1$
• C. $2$
• D. $4$

4. What is the order of $2$ in $\{1,2,\dots,6\}$ under multiplication modulo $7$?

• A. $2$
• B. $3$
• C. $6$
• D. $7$

5. Which of these groups must be abelian?

• A. Any group of order $6$
• B. The symmetry group of an equilateral triangle
• C. Any cyclic group
• D. Any group with an identity