User:MCPOliseno /Math850 HW1

1.1.25) Prove that if $$ x^2 \ $$ = 1 $$ \forall x \in G \ $$ then $$ G \ $$ is abelian.

Let $$ a, b \in G \ $$. Then $$ ab = a * 1 * b \ $$

= $$ a (ab)^2 b \ $$, since $$ x^2 \ $$ = 1

= $$ a(ab)(ab)b \ $$

= $$ (aa)(ba)(bb) \ $$

= $$ a^2(ba)b^2 \ $$

= $$ 1(ba)1 \ $$, since $$ x^2 \ $$ = 1

= $$ ba \ $$

Thus $$ ab = ba \ $$ and therefore $$ G \ $$ is abelian.

1.1.31) Prove that any finite group of $$ G \ $$ of even order contains an element of order 2.

Let $$ t(G) \ $$ = {$$ g\in G| g \ne g^{-1} \ $$}. We want to show that $$ t(G) \ $$ has an even number of elements and that every nonidentity element of $$ G - t(G) \ $$ has order 2.

Note that {$$ t(G) \ $$} = {$$ g_1, g_1^{-1} \ $$} $$ \cup \ $$ {$$ g_2, g_2^{-1} \ $$} $$ \cup \dots \ $$ $$ \cup \ $$ {$$ g_k, g_k^{-1} \ $$}. Then clearly $$ |t(G)| \ $$ is even.

Then $$ |G - t(G)| \ $$ = $$ |G| - |t(G)| \ $$, which is evidently even. Thus 2 divides $$ |G| \ $$ and 2 divides $$ |t(G)| \ $$ and therefore 2 divides $$ |G - t(G)| \ $$. Note that $$ 1 \in G \ $$ and $$ 1 \notin t(G) \ $$ which implies that $$ 1 \in \ $$ {$$ G - t(G) \ $$}. Thus every nonidentity element of $$ G - t(G) \ $$ has order 2.

1.6.8) Prove that is $$ n \ne m \ $$ then $$ S_n \ $$ and $$ S_m \ $$ are not isomorphic.

Finite groups of different order cannot be isomorphic. Thus $$ S_n \ $$ and $$ S_m \ $$ are not isomorphic.

1.6.9) Prove that $$ D_{24} \ $$ and $$ S_4 \ $$ are not isomorphic.

(In $$ D_{24} \exists \ $$ order 12 that is not in $$ S_4 \ $$)

2.1.13) Let $$ H \ $$ be a subgroup of the additive group of rational numbers with the property that $$ 1/x \in H \ $$ for every nonzero element $$ x \in H \ $$. Prove that $$ H = 0 \ $$ or $$ H = \Q \ $$.

Since, $$ H \ $$ is a subset of $$ \Q \ $$ $$ H \ $$ is closed under addition, then $$ \frac{a}{b} +\frac{a}{b} \dots + \frac{a}{b} \ $$ b times is still in $$ H \ $$. Thus $$ a \in H \implies \frac{1}{a} \in H \ $$. Similarly, $$ \frac{1}{a} + \frac{1}{a} + \dots + \frac{1}{a} \ $$ can be written a times, such that $$ 1 \in H \ $$ and therefore $$ \Z \subseteq H \ $$. Then $$ \forall x \in \Z, \frac{1}{a} \in H \implies H = \ $$ {$$ \frac{a}{b} : a, b \in \Z \ $$} which is $$ \Q \ $$ and thus $$ H = \Q \ $$.

2.1.17) Let $$ n \in \Z^+ \ $$ and let $$ F \ $$ be a field. Prove that the set $$ G \ $$ = {$$(a_{ij}) \in GL_n(F) | a_{ij} = 0 \forall i > j, a_{ii}=1 \forall i \ $$} is a subgroup of $$ GL_n(F) \ $$.

Let $$ A, B \in G \ $$. Then $$ A = \begin{bmatrix} 1 & a_{12} & \cdots & a_{1j} \\ 0 & 1 & \cdots & a_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \\ \end{bmatrix} \ $$ And $$ B = \begin{bmatrix} 1 & b_{12} & \cdots & b_{1j} \\ 0 & 1 & \cdots & b_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \\ \end{bmatrix} \ $$. Then $$ AB = \begin{bmatrix} 1 & c_{12} & \cdots & c_{1j} \\ 0 & 1 & \cdots & c_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \\ \end{bmatrix}\ $$ where $$ c_{ij} = a_{ij}b_{ij} \ $$. Thus $$ AB \in G \ $$. Now $$ A^{-1} \ $$ = $$ \begin{bmatrix} 1 & 1/a_{12} & \cdots & 1/a_{1j} \\ 0 & 1 & \cdots & 1/a_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \\ \end{bmatrix} \ $$. And thus $$ A^{-1} \in G \ $$ and therefore $$ G \ $$ is a subgroup of $$ GL_n(F) \ $$.

2.3.26) Let $$ \Z_n \ $$ be a cyclic group of order $$ n \ $$ and for each integer $$ a \ $$ $$ \sigma_a : \Z_n \to \Z_n \ $$ by $$ \sigma_a (x) = x^a \ $$ for all $$ x \in \Z_n \ $$.

(a) Prove that $$ \sigma_a \ $$ is an automorphism of $$ \Z_n \iff gcd(a,n) = 1 \ $$.

$$ \sigma_a \ $$ is an automorphism $$ \iff \ $$ it is surjective

$$ \iff \ $$ $$ f^{-1} (1) = x \ $$ $$ \iff \ $$ $$ ax = 1 \ $$ $$ \iff \ $$ $$ ax - 1 = bn \ $$ $$ \iff \ $$ $$ ax - bn = 1 \ $$ $$ \iff \ $$ $$ gcd (a,n) = 1 \ $$

(b) Prove that $$ \sigma_a = \sigma_b \iff a \equiv b(mod n) \ $$.

$$ \sigma_a = \sigma_b \iff ax = bx \ $$ $$ \iff \ $$ $$ ax-bx = 0 \ $$ $$ \iff \ $$ $$ x(a-b) = 0 \ $$ $$ \iff \ $$ $$ a-b = nz \ $$ for some $$ z \in \Z \ $$ $$ \iff \ $$ $$ a \equiv b(mod n) \ $$.

(c) Prove that every automorphism of $$ \Z_n \ $$is equal to $$ \sigma_a \ $$ for some integer $$ a\ $$.

(d) Prove that $$ \sigma_a \circ \sigma_b = \sigma_{ab} \ $$. Deduce that the map $$ \overline{a} \to \sigma_b \ $$ is an isomorphism of $$(\Z/n\Z)^x \ $$ onto the automorphism group of $$\Z_n \ $$.

2.4.14 c) A group $$ H \ $$ is called finitely generated if there is a finite set $$ A \ $$ such that $$ H =  \ $$. Prove that every finitely generated subgroup of the additive group $$ \Q \ $$ is cyclic.

Let $$ H \ $$ be finitely generated subgroup of $$ \Q \ $$. Then $$ H =  \ $$ where $$ a_i \in \Q, 1 \le i \le n \ $$. So $$ a_i = \frac{p_i}{q_i} \ $$ for $$ p_i, q_i \in \Z \ $$ and $$ 1 \le i \le n \ $$. We want to show that $$ H \subset <\frac{1}{k}> \ $$ where $$ k \ $$ is the product of the denominators of the $$ a_is \ $$/. So let $$ g \in H \ $$. Then $$ g = \alpha_1 a_1 + \dots + \alpha_n a_n \ $$ where $$ \alpha_i \in \Z, 1\le i \le n \ $$. Then $$ g = \alpha_1 \frac{p_1}{q_1} + \dots + \alpha_i \frac{p_i}{q_i} = \sum_{i=1}^{n} \alpha_i p_n (q_1 \dots q_n) / (q_1 \dots q_n) \ $$. Then $$ g \in <\frac{1}{k}> \ $$ where $$ k = q_1 \dots q_n \ $$. We know that any subgroup of a cyclic subgroup is cyclic. Therefore $$ H \ $$ is cyclic.