Abstract Algebra

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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 ${\displaystyle x^{2}=1\forall x\in G}$, 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 ${\displaystyle \emptyset \neq H\subseteq G}$ and ${\displaystyle \forall a,b\in H,ab^{-1}\in H}$, 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) ${\displaystyle \exists s\in G\,}$ 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 ${\displaystyle u,v,\in G,uN=vN\,}$ if and only if ${\displaystyle v^{-1}u\in N\,}$ 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 ${\displaystyle H\neq K}$, 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 (${\displaystyle ab)^{2}=a^{2}b^{2}}$ for all ${\displaystyle a,b\in G}$. Prove ${\displaystyle G}$ is abelian.

proof Let ${\displaystyle G}$ be a group such that ${\displaystyle (ab)^{i}=a^{i}b^{i}}$ 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.
• ${\displaystyle (a_{1}a_{2}a_{3}\cdot \cdot \cdot a_{n})^{-1}=a_{n}^{-1}a_{n-1}^{-1}a_{1}^{-1}\forall a_{1},a_{2},...,a_{n}\in G\,}$.
• If ${\displaystyle x\in G\,}$ then ${\displaystyle |x|=|x^{-1}|\,}$.
• If ${\displaystyle x\in G\,}$ then ${\displaystyle x^{2}=1\,}$ if and only if ${\displaystyle |x|=1\,\mathrm {or} \,2\,}$.
• If ${\displaystyle x^{2}=1\forall x\in G\,}$, 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 ${\displaystyle x\in D_{2n}\,}$ 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 ${\displaystyle n\geq 3\,}$.
• 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 ${\displaystyle S_{n}\,}$ 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 ${\displaystyle n\geq m\,}$ 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 ${\displaystyle \phi (g_{1}g_{2})=\phi (g_{1})\phi (g_{2})\forall g_{1},g_{2}\in G\,}$. This implies ${\displaystyle 1_{g}\mapsto 1_{H}\,}$ and ${\displaystyle \phi (g_{1}^{-1})\mapsto \phi (g_{1})^{-1}\,}$.
• ${\displaystyle Ker\phi =\{g\in G:\phi (g)=1_{H}\}\,}$
• ${\displaystyle Im\phi =\{h\in H:\exists g\in G\mathrm {s.t.} \phi (g)=h\}\,}$
• 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 ${\displaystyle N\in \mathbb {N} \,}$ 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 ${\displaystyle \exists x\in R\,}$ 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 ${\displaystyle m\in \mathbb {Z} ^{+}\,}$. 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 ${\displaystyle m\in \mathbb {Z} ^{+}\,}$. Prove that ${\displaystyle rx\,}$ is nilpotent for all ${\displaystyle r\in R\,}$.

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

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