Absract Algebra

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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 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.

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=<a>\, 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 <a>\cap<b>={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.

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)\,

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