1. ## Groups etc

Question 13.
(1) Suppose f : G -> H is a homomorphism, and a is an element G has order n. Show that the
order of f(a) must divide n.
(2) Let G be any finite group. Find all homomorphisms f :G->Z.
(3) Prove that (R \ {0}, ·) is not isomorphic to (R, +).

Question 17
(1) Find the order of A = (−1 −1 ) in GL2(R).
( 1 0 )

(2) Find the order of (2, 8,−1) in (Z/9Z)× × Z/28Z × (Z \ {0}). Justify your answer.

Question 21

(1) Define
f : R \ {0} -> R\ {0} by f(x) = x2 (the group operation is multiplication).
Prove that f is a homomorphism and give the kernel. Is f an isomorphism?
(2) Prove that {2k | k 2 Z} is a subgroup of (R \ {0}, ·).

2. Originally Posted by asw-88
Question 13.
(1) Suppose f : G -> H is a homomorphism, and a is an element G has order n. Show that the
order of f(a) must divide n.
(2) Let G be any finite group. Find all homomorphisms f :G->Z.
(3) Prove that (R \ {0}, ·) is not isomorphic to (R, +).
Let $\displaystyle a^k = e$ where $\displaystyle k$ is the order of $\displaystyle a$. Then $\displaystyle \phi (a^k) = \phi(e) = e'$ by homomorphisms it means $\displaystyle (\phi (a))^k = e'$. This means the order of $\displaystyle \phi(a)$ must divide $\displaystyle k$ (because if $\displaystyle n$ is the order and if $\displaystyle (\phi(a))^m = e'$ then $\displaystyle n|m$ by properties of element orders).

(2) Let G be any finite group. Find all homomorphisms f :G->Z.
By above exercise if $\displaystyle a\in G$ and $\displaystyle a$ has finite order (which it does for the group is finite) it must mean $\displaystyle \phi(a)$ has finite order. But the only element in $\displaystyle (\mathbb{Z},+)$ which has finite order is $\displaystyle 0$. Thus the homomorphism is the trivial one, i.e. $\displaystyle \phi (a) = 0$ for all $\displaystyle a\in G$.

(3) Prove that (R \ {0}, ·) is not isomorphic to (R, +)
If $\displaystyle (\mathbb{R}^{\text{x}},\cdot) = (\mathbb{R}, +)$ then they both would have the same number of finite order elements. Note, $\displaystyle \mathbb{R}^{\text{x}}$ has two elements while $\displaystyle \mathbb{R}$ has only one.

3. Originally Posted by asw-88
Question 17
(1) Find the order of A = (−1 −1 ) in GL2(R).
( 1 0 )
You need to find the smallest positive integer $\displaystyle k$ such that $\displaystyle A=\left( \begin{array}{cc}-1&-1\\1&0 \end{array} \right)^k = \text{I} = \left(\begin{array}{cc}1&0\\0&1\end{array} \right)$.

(2) Find the order of (2, 8,−1) in (Z/9Z)× × Z/28Z × (Z \ {0}). Justify your answer.
The order of $\displaystyle [8]_{28}$ in $\displaystyle \mathbb{Z}_{28}$ is infinite. Check your problem again.