Abstract Algebra

Abstract algebra in the Encyclopedia.

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 $x^{2}=1\forall x\in G$, 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 $\emptyset \neq H\subseteq G$ and $\forall a,b\in H,ab^{{-1}}\in H$, 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) $\exists s\in G\,$ 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 $u,v,\in G,uN=vN\,$ if and only if $v^{{-1}}u\in N\,$ 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 $H\neq K$, prove $H\cap K=$.

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

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

proof Let $G$ be a group such that $(ab)^{i}=a^{i}b^{i}$ 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.
• $(a_{1}a_{2}a_{3}\cdot \cdot \cdot a_{n})^{{-1}}=a_{n}^{{-1}}a_{{n-1}}^{{-1}}a_{1}^{{-1}}\forall a_{1},a_{2},...,a_{n}\in G\,$.
• If $x\in G\,$ then $|x|=|x^{{-1}}|\,$.
• If $x\in G\,$ then $x^{2}=1\,$ if and only if $|x|=1\,{\mathrm {or}}\,2\,$.
• If $x^{2}=1\forall x\in G\,$, then $G\,$ is abelian.
• $A\times B\,$ is abelian if and only if both $A\,$ and $B\,$ are abelian.

Dihedral Groups

• If $x\in D_{{2n}}\,$ 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 $n\geq 3\,$.
• 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 $n\geq m\,$ then the number of $m\,$-cycles in $S_{n}\,$ is given by ${\frac {n(n-1)(n-2)...(n-m+1)}{m}}\,$.

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 $\phi (g_{1}g_{2})=\phi (g_{1})\phi (g_{2})\forall g_{1},g_{2}\in G\,$. This implies $1_{g}\mapsto 1_{H}\,$ and $\phi (g_{1}^{{-1}})\mapsto \phi (g_{1})^{{-1}}\,$.
• $Ker\phi =\{g\in G:\phi (g)=1_{H}\}\,$
• $Im\phi =\{h\in H:\exists g\in G{\mathrm {s.t.}}\phi (g)=h\}\,$
• 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 $N\in {\mathbb {N}}\,$ 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 $\exists x\in R\,$ 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 $m\in {\mathbb {Z}}^{+}\,$. 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 $m\in {\mathbb {Z}}^{+}\,$. Prove that $rx\,$ is nilpotent for all $r\in R\,$.

solution Let $x\,$ be a nilpotent element of the commutative ring $R\,$. Let $x^{m}=0\,$ for minimal $m\in {\mathbb {Z}}^{+}\,$. Deduce that the sum of nilpotent element and a unit is a unit.
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 $1\neq 0\,$.

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 $i,j\in \left\{1,2,...,k\right\}\,$ with $i\neq j\,$ 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 $a_{1},...,a_{k}\in {\mathbb {Z}}\,$ there is a solution $x\in {\mathbb {Z}}\,$ 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

$x=a_{1}t_{1}n_{1}'+a_{2}t_{2}n_{2}'+\cdot \cdot \cdot +a_{k}t_{k}n_{k}'\mod n\,$.

(c) Solve the simultaneous systems of congruences

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

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 ${\mathbb {Z}}[i]^{\times }$

Fields

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

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

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

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