Abstract Algebra

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Abstract algebra in the Encyclopedia.

Contents


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 G1 is a subgroup of G and H1 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 pq is cyclic.

proof Let G be a group such that (ab)2 = a2b2 for all a,b\in G. Prove G is abelian.

proof Let G be a group such that (ab)i = aibi 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}\,.


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.
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\,.



(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}.

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