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 ({\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=<e>.

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}}.