# Abstract Algebra

Abstract algebra in the Encyclopedia.

## Basic Stuff

solution Explain the "null set."

## Groups

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

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

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

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

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

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

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

proof If $\displaystyle G_1$ is a subgroup of $\displaystyle G$ and $\displaystyle H_1$ is a subgroup of $\displaystyle H$ , prove $\displaystyle G1\times H1$ is a subgroup of $\displaystyle G\times H$ .

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

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

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

proof Let $\displaystyle G$ be a group such that $\displaystyle (ab)^i=a^ib^i$ for all $\displaystyle a,b\in G$ and for three consecutive integers $\displaystyle i$ . Prove $\displaystyle G$ 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 $\displaystyle \mathbb{Z}/n\mathbb{Z}\,$ is associative.
• Multiplication of residue classes in $\displaystyle \mathbb{Z}/n\mathbb{Z}\,$ is associative.
• A finite group is abelian if and only if its group table is a symmetric matrix.
• $\displaystyle (a_1 a_2 a_3 \cdot\cdot\cdot a_n)^{-1} = a_n^{-1} a_{n-1}^{-1} a_1^{-1} \forall a_1, a_2, ..., a_n \isin G\,$ .
• If $\displaystyle x \isin G\,$ then $\displaystyle |x|=|x^{-1}|\,$ .
• If $\displaystyle x \isin G\,$ then $\displaystyle x^2=1\,$ if and only if $\displaystyle |x|=1 \,\mathrm{ or }\, 2\,$ .
• If $\displaystyle x^2=1 \forall x \isin G\,$ , then $\displaystyle G\,$ is abelian.
• $\displaystyle A \times B\,$ is abelian if and only if both $\displaystyle A\,$ and $\displaystyle B\,$ are abelian.

Dihedral Groups

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

Symmetric Groups

• $\displaystyle S_n\,$ is a non-abelian group for all $\displaystyle n\ge 3\,$ .
• 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 $\displaystyle S_n\,$ if and only if its cycle decomposition is a product of commuting 2-cycles.
• Let $\displaystyle p\,$ be a prime. Show that an element has order $\displaystyle p\,$ in $\displaystyle S_n\,$ if and only if its cycle decomposition is a product of commuting p-cycles.
• If $\displaystyle n\ge m\,$ then the number of $\displaystyle m\,$ -cycles in $\displaystyle S_n\,$ is given by $\displaystyle \frac{n(n-1)(n-2)...(n-m+1)}{m}\,$ .

Matrix Groups

• $\displaystyle GL_n(F)=\{A|A \,$ is an $\displaystyle n\times n \,$ matrix with entries from $\displaystyle F \,$ and nonzero determinant.
• $\displaystyle |GL_n(\mathbb{F}_2)|=6\,$
• $\displaystyle GL_n(\mathbb{F}_2)\,$ is non-abelian.

Homomorphisms and Isomorphisms

• $\displaystyle \phi : G \to H\,$ is a homomorphism if $\displaystyle \phi(g_1 g_2)=\phi(g_1)\phi(g_2) \forall g_1,g_2\isin G\,$ . This implies $\displaystyle 1_g \mapsto 1_H\,$ and $\displaystyle \phi(g_1^{-1})\mapsto\phi(g_1)^{-1}\,$ .
• $\displaystyle Ker \phi = \{g\isin G: \phi(g)=1_H\}\,$
• $\displaystyle Im \phi = \{h\isin H: \exists g\isin G \mathrm{s.t.} \phi(g)=h\}\,$
• The exponential map $\displaystyle \exp: \mathbb{R}\to\mathbb{R}^+\,$ defined by $\displaystyle \exp(x)=e^x\,$ is an isomorphism from $\displaystyle (\mathbb{R},+)\,$ to $\displaystyle (\mathbb{R}^+,\times)\,$ .

## Rings

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

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

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

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

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

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

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

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

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

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

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

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

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

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

solution Let $\displaystyle x\,$ be a nilpotent element of the commutative ring $\displaystyle R\,$ . Let $\displaystyle x^m=0\,$ for minimal $\displaystyle m\isin \mathbb{Z}^+\,$ . Deduce that the sum of nilpotent element and a unit is a unit.
solution Prove that the center $\displaystyle \mathbb{Z}\,$ of a ring $\displaystyle R\,$ 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 $\displaystyle R\,$ be a ring with identity $\displaystyle 1\ne 0\,$ .

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

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

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

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

$\displaystyle x\equiv a_1 \mod n_1$

$\displaystyle x\equiv a_2 \mod n_2$

$\displaystyle ...\,$

$\displaystyle x\equiv a_k \mod n_k$

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

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

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

(c) Solve the simultaneous systems of congruences

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

and

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

## Euclidean Domains

solution Define a Euclidean Domain.

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

## Fields

solution Calculate the splitting field $\displaystyle E\,$ of $\displaystyle f(x)=x^3-5\,$ over $\displaystyle \mathbb{Q}$ . What is $\displaystyle [E:\mathbb{Q}]$ ?

solution Prove that the polynomial is irreducible: $\displaystyle x^6+30x^5-15x^3+6x-120\,$

solution Prove that the polynomial is irreducible: $\displaystyle x^4+4x^3+6x^2+2x+1\,$

solution Show that the splitting field of $\displaystyle f(x)=x^4+1\in\mathbb{Q}[x]$ is a simple extension of $\displaystyle \mathbb{Q}$ .