Abstract Algebra

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Abstract algebra in the Encyclopedia.

1 Basic Stuff
2 Groups
3 Groups- facts and examples
4 Rings
5 Chinese Remainder Theorem
6 Euclidean Domains
7 Fields

ABSTRACT ALGEBRA BOOKS

Basic Stuff

solution Explain the "null set."

Groups

proof Prove that the additive groups $(\mathbb{R}^n,+)$ and $(\mathbb{R}^m,+)$ are isomorphic for $n, m \in \mathbb{N}$

proof Prove that if $x^2=1\forall x\isin G$ , then $G$ is abelian.

proof Prove that if $G$ is a group and $H, K$ are subgroups of $G$ , then $H\cap K$ is a group.

proof Prove that $\empty \ne H\subseteq G$ and $\forall a,b\isin H, ab^{-1}\isin H$ , if and only if $H\,$ is a subgroup of $G\,$ .

proof Prove that a group homomorphism maps the identity to the identity and inverses to inverses.

proof Prove that the kernel of a homomorphism from a group is a subgroup of that group.

solution Define actions, centralizers, normalizers, stabilizers, and centers.

proof Prove that if $G=\langle a\rangle\,$ and $|G|=n\,$ then the following are equivalent:
   (a) $|a^r|=n\,$ which means $a^r\,$ is a generator of $G\,$ .
   (b) $(r,n)=1\,$ i.e. $r$ and $n\,$ are relatively prime.
   (c) $\exists s\isin G\,$ such that $rs\equiv 1(\mathrm{mod} \,\,n)\,$ .

solution Let $a\,$ and $b\,$ belong to the group $G\,$ . If $ab=ba\,$ and $|a|=m, |b|=n\,$ , where $m\,$ and $n\,$ are relatively prime, show that $|ab|=mn\,$ and that $\langle a\rangle\cap\langle b\rangle={1}\,$ .

proof Let $N\,$ be any subgroup of the group $G\,$ . The set of left cosets of $N\,$ in $G\,$ form a partition of $G\,$ . Furthermore, for all $u,v,\isin G, u N = v N\,$ if and only if $v^{-1}u\isin N\,$ and in particular, $uN=vN\,$ if and only if $u\,$ and $v\,$ are representatives of the same coset.

proof If $G 1$ is a subgroup of $G$ and $H 1$ is a subgroup of $H$ , prove $G1\times H1$ is a subgroup of $G\times H$ .

proof Let $H$ and $K$ , each of prime order $p$ , be subgroups of a group $G$ . If $H\ne K$ , prove $H\cap K=<e>$ .

proof If $p$ and $q$ are prime, show every proper subgroup of a group of order $p q$ is cyclic.

proof Let $G$ be a group such that ( $a b) 2 = a 2 b 2$ for all $a,b\in G$ . Prove $G$ is abelian.

proof Let $G$ be a group such that $(a b) i = a i b i$ for all $a,b\in G$ and for three consecutive integers $i$ . Prove $G$ is abelian.

proof Prove that a group of order 56 has a normal Sylow p-subgroup for some prime dividing its order.

proof Prove that a group of order 312 has a normal Sylow p-subgroup.

Groups- facts and examples

Addition of residue classes in $\mathbb{Z}/n\mathbb{Z}\,$ is associative.

Multiplication of residue classes in $\mathbb{Z}/n\mathbb{Z}\,$ is associative.

A finite group is abelian if and only if its group table is a symmetric matrix.

$(a_1 a_2 a_3 \cdot\cdot\cdot a_n)^{-1} = a_n^{-1} a_{n-1}^{-1} a_1^{-1} \forall a_1, a_2, ..., a_n \isin G\,$ .

If $x \isin G\,$ then $|x|=|x^{-1}|\,$ .

If $x \isin G\,$ then $x^2=1\,$ if and only if $|x|=1 \,\mathrm{ or }\, 2\,$ .

If $x^2=1 \forall x \isin G\,$ , then $G\,$ is abelian.

$A \times B\,$ is abelian if and only if both $A\,$ and $B\,$ are abelian.

Dihedral Groups

If $x \isin D_{2n}\,$ and $x\,$ is not a power of $r\,$ , then $rx=xr^{-1}\,$ .

Every element of $D_{2n}\,$ which is not a power of $r\,$ is of order 2.

Symmetric Groups

$S_n\,$ is a non-abelian group for all $n\ge 3\,$ .

Disjoint cycles commute.

The order of a permutation is the l.c.m. of the lengths of the cycles in its cycle decomposition.

An element has order 2 in $S_n\,$ if and only if its cycle decomposition is a product of commuting 2-cycles.

Let $p\,$ be a prime. Show that an element has order $p\,$ in $S_n\,$ if and only if its cycle decomposition is a product of commuting p-cycles.

If $n\ge m\,$ then the number of $m\,$ -cycles in $S_n\,$ is given by $\frac{n(n-1)(n-2)...(n-m+1)}{m}\,$ .

ABSTRACT ALGEBRA BOOKS

Matrix Groups

$GL_n(F)=\{A|A \,$ is an $n\times n \,$ matrix with entries from $F \,$ and nonzero determinant.

$|GL_n(\mathbb{F}_2)|=6\,$

$GL_n(\mathbb{F}_2)\,$ is non-abelian.

Homomorphisms and Isomorphisms

$\phi : G \to H\,$ is a homomorphism if $\phi(g_1 g_2)=\phi(g_1)\phi(g_2) \forall g_1,g_2\isin G\,$ . This implies $1_g \mapsto 1_H\,$ and $\phi(g_1^{-1})\mapsto\phi(g_1)^{-1}\,$ .

$Ker \phi = \{g\isin G: \phi(g)=1_H\}\,$

$Im \phi = \{h\isin H: \exists g\isin G \mathrm{s.t.} \phi(g)=h\}\,$

The exponential map $\exp: \mathbb{R}\to\mathbb{R}^+\,$ defined by $\exp(x)=e^x\,$ is an isomorphism from $(\mathbb{R},+)\,$ to $(\mathbb{R}^+,\times)\,$ .

Rings

solution In the ring $R=\mathbb{Z}\,$ , prove the following:

(a) $(m) \supset (n)\,$ if and only if $m\big| n\,$

(b) $(m,n) = (m)+(n)=(d)\,$ where $d=\gcd(m,n)\,$ . You need to show both equalities. Note that this implies that any ideal in the ring $\mathbb{Z}\,$ is principal.

solution Determine whether $x^4+15x+7\,$ is irreducible over $\mathbb{Q}\,$ or not.

solution Find a g.c.d. of $6+7i\,$ and $12-3i\,$ in $\mathbb{Z}[i]\,$ by the Euclidean algorithm.

solution Show that $1+3\sqrt{-5}\,$ is irreducible but not a prime in $\mathbb{Z}[\sqrt{-5}]\,$ .

solution Show that $(x-5)\,$ is a maximal ideal of $\mathbb{C}[x]\,$ .

solution (Dis)prove: Let $R\,$ be a commutative ring with more than one element. If for every nonzero element $a\,$ of $R\,$ , we have $aR=R\,$ , then $R\,$ is a field.

solution (Dis)prove: $\mathbb{Z}/(p)\times\mathbb{Z}/(p)\cong \mathbb{Z}/(p^2)\,$ as rings, where $p\,$ is prime.

solution Show that an integral domain $R\,$ with a descending chain condition (if $I_1\supseteq I_2\supseteq I_3\supseteq\cdot\cdot\cdot\,$ is a descending chain of ideals, then there exists $N\isin\mathbb{N}\,$ such that $I_N = I_{N+1} = \cdot\cdot\cdot\,$ ) is a field.

proof Let $R$ be an integral domain. Suppose that existence of factorizations holds in $R$ . Prove that $R$ is a unique factorization domain if and only if every irreducible element is prime.

solution Prove: If $R\,$ is an integral domain and $\exists x \isin R\,$ s.t. $x^2=1\,$ then $x=\pm 1\,$ .

solution Let $x\,$ be a nilpotent element of the commutative ring $R\,$ . Let $x^m=0\,$ for minimal $m\isin \mathbb{Z}^+\,$ . Prove that $x\,$ is either zero or a zero divisor.

solution Let $x\,$ be a nilpotent element of the commutative ring $R\,$ . Let $x^m=0\,$ for minimal $m\isin \mathbb{Z}^+\,$ . Prove that $rx\,$ is nilpotent for all $r\isin R\,$ .

solution Let $x\,$ be a nilpotent element of the commutative ring $R\,$ . Let $x^m=0\,$ for minimal $m\isin \mathbb{Z}^+\,$ . Deduce that the sum of nilpotent element and a unit is a unit.

ABSTRACT ALGEBRA BOOKS

solution Prove that the center $\mathbb{Z}\,$ of a ring $R\,$ is a subring containing the identity.

solution Prove that the center of a division ring is a field.

Chinese Remainder Theorem

Statement of theorem: Let $R\,$ be a ring with identity $1\ne 0\,$ .

Let $A_1,A_2,...,A_k\,$ be ideals of $R\,$ . The map $R\rightarrow R/A_1 \times R/A_2 \times ... \times R/A_k\,$ defined by $r\mapsto (r+A_1,r+A_2,...,r+A_k)\,$ is a ring homomorphism with kernel $A_1 \cap A_2 \cap \cdot\cdot\cdot \cap A_k\,$ . If for each $i, j \isin \left\{1,2,...,k\right\}\,$ with $i\ne j\,$ the ideals $A_i\,$ and $A_j\,$ are comaximal, then this map is surjective and $A_1\cap A_2\cap \cdot\cdot\cdot \cap A_k = A_1 A_2 \cdot\cdot\cdot A_k\,$ , so

$R/(A_1 A_2 \cdot\cdot\cdot A_k) = R/(A_1\cap A_2\cap \cdot\cdot\cdot \cap A_k)\,$ $\cong R/A_1 \times R/A_2 \times \cdot\cdot\cdot R/A_k\,$

solution Let $n_1,n_2,...,n_k\,$ be integers which are coprime to each other.

(a) Show that the Chinese Remainder Theorem implies that for any $a_1, ..., a_k\isin\mathbb{Z}\,$ there is a solution $x\isin \mathbb{Z}\,$ to the simultaneous congruences

$x\equiv a_1 \mod n_1$

$x\equiv a_2 \mod n_2$

$...\,$

$x\equiv a_k \mod n_k$

and that the solution $x$ is unique mod $n = n_1 n_2 \cdot\cdot\cdot n_k\,$ .

(b) Let $n_i' =n/n_i\,$ be the quotient of $n\,$ by $n_i\,$ . Prove that the solution $x\,$ in (a) is given by

$x=a_1 t_1 n_1' + a_2 t_2 n_2' + \cdot\cdot\cdot + a_k t_k n_k' \mod n\,$ .

$x\equiv 1\mod 8, x\equiv 2\mod 25, x\equiv 3\mod 81\,$

and

$y\equiv 5\mod 8, y\equiv 12\mod 25, y\equiv 47\mod 81\,$ .

Euclidean Domains

solution Define a Euclidean Domain.

solution Let R be a Euclidean Domain with a function $\varphi$ . Prove that
(a) $\varphi(1) = \min\{\varphi(a) \ |\ a \in R \backslash \{0\}\}$
(b) $R^\times = \{r \in R \backslash \{0\} \ |\ \varphi(r) = \varphi(1)\}$
(c) Use (b) to determine $\mathbb{Z}^\times, F^\times, F[X]^\times,$ and $\mathbb{Z}[i]^\times$