## Analogy between quotient groups and quotient topology

The quotient topology is the unique topology on the set \$X/{sim}\$ of equivalence classes such that for any topological space …

## Alternative proof that \$U(n^2-1)\$ is not cyclic for \$n>2\$.

Using Theorem 4.4, one sees that if \$2\$ divides \$phi(n^2-1)\$ (which is the order of \$U(n^2-1)\$, an even number) and …

## Alternative proof of : \$n = sum_{d|n} phi(d)\$ which is the sum of Euler Phi Function over Divisors

The idea is you want to count the elements in \$mathbb{Z}_{n}\$ in two ways. First, we note that \$|mathbb{Z}_{n}| = …

## Almost normal subgroups: Is there any notion which is weaker than normal subgroup?

Consider the set of all conjugates of a subgroup \$Hleq G\$, defined by \$C={gHg^{-1}mid gin G}\$. If \$H\$ is normal, …

## Algebraic structure for the floating-point arithmetic.

Look into D.E. Knuth’s The Art of Computer Programming, Vol. 2, Section 4.2.2 A, p. 214-223, he gives an axiomatic …

## A group with no proper non-trivial subgroups

Given \$ain G\$, \$langle arangle\$ is defined to be the smallest subgroup of \$G\$ containing \$a\$ as an element. Knowing …

## A group with 4 elements

You’ve already been given most of the entries for the group’s Cayley table, so let’s just see if we can …

## A group of order \$n^2\$ with \$n+1\$ subgroups of order \$n\$ with trivial intersection is abelian

Let \$H,K\$ be two distinct subgroup of order \$n\$. Now \$mid HKmid=frac{mid Hmid mid Kmid}{mid Hcap Kmid}=mid Hmid mid Kmid=n^2=mid …

## A group of order \$30\$ has a normal \$5\$-Sylow subgroup.

Short Answers: 1.- Because \$,P_5le N_G(P_5),\$ and \$,|N_G(P_5)|=|P_5|,\$ 2.- The “2nd Noether Theorem” seems to be what others (like me) …

## A group of order 20

Suppose, by way of contradiction, that there is a normal \$2\$-Sylow subgroup \$D\$. If \$D\$ is cyclic, then it has …