Isomorphism of groups Quiz

1. If $\phi:G\to H$ is an isomorphism, what is $\phi(e_G)$?

• A. Any element of $H$
• B. The identity of $H$
• C. $\phi(a)^{-1}$
• D. $0$

2. $G$ and $H$ both have order $8$. $G$ has three elements of order $2$ and $H$ has one. Are they isomorphic?

• A. Yes, because they have the same order.
• B. Yes, because both have an identity.
• C. No, because an isomorphism preserves the number of elements of each order.
• D. No, because $G$ and $H$ use different operations.

3. Which pair of groups cannot be isomorphic?

• A. $\{1,3,5,7\}$ modulo $8$, and $\{0,1,2,3\}$ under $+_4$
• B. $\{0,1,2,3\}$ under $+_4$, and $\{1,i,-1,-i\}$
• C. $\{0,\dots,5\}$ under $+_6$, and $\{1,\dots,6\}$ modulo $7$
• D. $\{0,1\}$ under $+_2$, and $\{1,-1\}$ under multiplication

4. $\phi(k)=3^k\pmod7$ maps $\{0,\dots,5\}$ under $+_6$ to $\{1,\dots,6\}$ under multiplication modulo $7$. Find $\phi(4)$.

• A. $2$
• B. $3$
• C. $5$
• D. $4$

5. Which is not necessarily preserved by an isomorphism?

• A. The number of elements of order $2$
• B. The labels of the elements
• C. Being abelian
• D. The order of the group