# Absract Algebra

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## 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 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.

proof Prove that if $\displaystyle x^2=1\forall x\isin G$ , 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 $\displaystyle \empty \ne H\subseteq G$ and $\displaystyle \forall a,b\isin H, ab^{-1}\isin H$ , 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 $\displaystyle G=\,$ 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) $\displaystyle \exists s\isin G\,$ 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 $\displaystyle \cap={1}\,$ .

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 $\displaystyle u,v,\isin G, u N = v N\,$ if and only if $\displaystyle v^{-1}u\isin N\,$ and in particular, $\displaystyle uN=vN\,$ if and only if $\displaystyle u\,$ and $\displaystyle v\,$ are representatives of the same coset.

## 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.
• $\displaystyle (a_1 a_2 a_3 \cdot\cdot\cdot a_n)^{-1} = a_n^{-1} a_{n-1}^{-1} a_1^{-1} \forall a_1, a_2, ..., a_n \isin G\,$ .
• If $\displaystyle x \isin G\,$ then $\displaystyle |x|=|x^{-1}|\,$ .
• If $\displaystyle x \isin G\,$ then $\displaystyle x^2=1\,$ if and only if $\displaystyle |x|=1 \,\mathrm{ or }\, 2\,$ .
• If $\displaystyle x^2=1 \forall x \isin G\,$ , 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 $\displaystyle x \isin D_{2n}\,$ 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 $\displaystyle n\ge 3\,$ .
• 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 $\displaystyle S_n\,$ 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 $\displaystyle n\ge m\,$ 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 $\displaystyle \phi(g_1 g_2)=\phi(g_1)\phi(g_2) \forall g_1,g_2\isin G\,$ . This implies $\displaystyle 1_g \mapsto 1_H\,$ and $\displaystyle \phi(g_1^{-1})\mapsto\phi(g_1)^{-1}\,$ .
• $\displaystyle Ker \phi = \{g\isin G: \phi(g)=1_H\}\,$
• $\displaystyle Im \phi = \{h\isin H: \exists g\isin G \mathrm{s.t.} \phi(g)=h\}\,$
• 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 $\displaystyle (\mathbb{R}^+,\times)\,$