Articles of abstract algebra

Decomposition Theorem for Posets

There is in module theory the following (krull-shmidt) theorem: If $(M_i)_{i\in I}$ and $(N_j)_{j\in J}$ are two families of simple module such that $\bigoplus\limits_{i\in I} M_i \simeq \bigoplus\limits_{j\in J} M_j$, then there exists a bijection $\sigma:I\rightarrow J$ such that $M_i \simeq N_{\sigma(i)}$. Is there a similar kind of theorem for partially ordered sets? More precisely, […]

Proof that $\mathbb{R}$ is not a finite dimensional vector space

How can we prove that the vector space of polynomials in one variable, $\mathbb{R}[x]$ is not finite dimensional?

Finding the kernel of ring homomorphisms from rings of multivariate polynomials

I am trying to find the kernels of the following ring homomorphisms: $$ f:\Bbb C[x,y]\rightarrow\Bbb C[t];\ f(a)=a\ (a\in\Bbb C),f(x)=t^2,f(y)=t^5. $$ $$ g:\Bbb C[x,y,z]\rightarrow\Bbb C[t,s];\ g(a) = a\ (a\in\Bbb C), g(x)=t^2,g(y)=ts,g(z)=s^2. $$ $$ h:\Bbb C[x,y,z]\rightarrow\Bbb C[t];\ h(a)=a\ (a\in\Bbb C), h(x)=t^2, h(y)=t^3, h(z)=t^4. $$ I want to write them as ideals generated by as few elements as […]

Weyl group, permutation group

Let $U(n)$ be the unitary group and $T$ its maximal torus (group of diagonal matrix) and $N(T)$ the normalizer of $T$ in $G$. Why $N(T)/T$ is the permutation group $S_{n}$?

Has $S$ infinitely many nilpotent elements?

Let $S$ be a ring with identity (but not necessarily commutative) and $f:M_{2}(\mathbb R)→S$ a non zero ring homomorphism ($M_{2}(\mathbb R)$ is the ring of all $2\times 2$ matrices). Has $S$ infinitely many nilpotent elements?

The degree of the extension $F(a,b)$, if the degrees of $F(a)$ and $F(b)$ are relatively primes.

Let $E$ be an extension of $F$, and let $a, b \in E$ be algebraic over $F$. Suppose that the extensions $F(a)$ and $F(b)$ of $F$ are of degrees $m$ and $n$, respectively, where $(m,n)=1$. Show that $[F(a,b):F]=mn$. Since $[F(a,b):F]=[F(a,b):F(a)][F(a):F]$ and $[F(a):F]=n$ we have $n|[F(a,b):F]$ with the same argument we prove that $m|[F(a,b):F]$, then $mn|[F(a,b):F]$ […]

Showing that the only units in $\mathbb{Z}$ are $1,\, -1, \, i, \, -i$?

How would you show that the only units in $\mathbb{Z}[i] :=\{a + ib \, |\, a,b\in \mathbb{Z}\}$ are $1,\, -1, \, i, \, -i$?

Enumerating Sylow $2$-subgroups of Dihedral Group (of order $2^{\alpha}k$ for $k$ odd).

Let $2n = 2^a k$ for $k$ odd. Prove that the number of Sylow $2$-subgroups of $D_{2n}$ is $k$. I managed to prove this result by showing that the normalizer of any Sylow $2$-subgroup is itself. The result immediately follows, in that case. However, my problem is that I came across a different solution to […]

Proving a set is an abelian group.

I am trying to prove that $(G, *)$ is an abelian group with $G=(-1,1)$ and $a*b=$$\frac{x+y}{1+xy}$. Thus far I have found that the identity element $e=0$. From here, I set $a*b=0$ and found $a^{-1}$ to be $-a$. My work for trying to prove closure and that the set is abelian is: Let $a,b \in G$, […]

How far can we go with group isomorphisms?

The following is quoted from Wikipedia: From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished. But this statement is too general. What I’m wondering about is whether there is a limit to these properties for which isomorphic groups “need not be distinguished”. For instance, I have been […]