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\leftline{Mathematics 243 -- Algebraic Structures}
\leftline{Definitions Quiz 9 -- November 30, 2012}
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\leftline{Your Name: $\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$}
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A) (5) Let $a$ be an element of a group $G$.  Define: the cyclic subgroup of $G$ defined by $a$, written
$\langle a \rangle$.   

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B) (5)  Let $\phi : ({\bf R}, +) \to ({\bf R}_+, \cdot)$ be the mapping defined by $\phi(x) = 7^x$.  Is
$\phi$ a group homomorphism?  Why or why not?   (Note: ${\bf R}_+ = \{y \in {\bf R} \mid y > 0\}$, which 
is a group under multiplication.) 

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