Order, subgroups and Lagrange's theorem Quiz

1. A group has order 18. Which cannot be the order of a subgroup?

• A. 6
• B. 9
• C. 12
• D. 3

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

• A. $3$
• B. $2$
• C. $6$
• D. $4$

3. An element of a group of order 20 has order 6. Is this possible?

• A. Yes, 6 divides 20
• B. Yes, any order up to 20 is possible
• C. No, 6 does not divide 20
• D. Only if the group is cyclic

4. A group has order 7. Which statement is true?

• A. It must have a subgroup of order 3
• B. It has exactly 3 subgroups
• C. It is not cyclic
• D. It is cyclic

5. Which is a subgroup of $\{0,1,2,3,4,5\}$ under addition modulo 6?

• A. $\{0,2,4\}$
• B. $\{0,1,2\}$
• C. $\{1,3,5\}$
• D. $\{0,3,4\}$