Intereting Posts

zeros of a polynomial
We have sums, series and integrals. What's next?
How to prove a set of positive semi definite matrices forms a convex set?
Axiom Systems and Formal Systems
Convergence/Divergence of $\int_{0}^{1/e} \frac{\log \left(\frac{1}{x}\right)}{(\log^2 (x)-1)^{3/2}} \mathrm{dx}$
Can the integers be made into a vector space over any Finite Field?
How many words can be formed using all the letters of “DAUGHTER” so that vowels always come together?
Correct scaling factor in Lagrange's formula for triple cross product
Continuity from below and above
Can you check my proof on the characterization of the trace function?
Uniform distribution on unit disk
The Degree of Zero Polynomial.
Prove or disprove that $\exists a,b,c\in\mathbb{Z}_+\ \forall n\in\mathbb{Z}_+\ \exists k\in\mathbb{Z}_+\colon\ a^k+b^k+c^k = 0(\mathrm{mod}\ 2^n)$
EigenValues and EigenVectors in PCA?
Quadratic reciprocity and proving a number is a primitive root

I am told some information about a group $G$ of order $168$. All we are told about G is that: It has one element of order one, $21$ elements of order $2$, $56$ elements of order $3$, $42$ elements of order $4$ and $48$ elements of order $7$. and later it will be proved to be simple. But for now I am asked:

By looking at the possible orders of subgroups $N$ of $G$, show that if $N$ is a non-trivial proper normal subgroup of $G$, then $|N| ∈ \{2, 4\}$ by using the following fact: For $N$, a normal subgroup of index $n$ of $G$, let $a$ be an element of order $m$. Then if gcd$(m,n)=1$ then $a\in N$.

So at the moment we are not using the fact that $G$ is simple, this will be proved later on.

- $X$ is a basis for free abelian group $A_{n}$ if and only if $\det (M) = \pm 1$
- Find the number of $n$ by $n$ matrices conjugate to a diagonal matrix
- Is this an equivalent statement to the Fundamental Theorem of Algebra?
- Factoring polynomials of the form $1+x+\cdots +x^{p-1}$ in finite field
- If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative.
- Showing that $\mathbb{Q}$ contains multiplicative inverses

So I have tried using the fact that $168=|N||G:N|$. I also know that $|G:N|=|G/N|=n$ but don’t know where to go from here.

- Why can't the Polynomial Ring be a Field?
- Transcendental number
- $F/(x^2)\cong F/(x^2 - 1)$ if and only if F has characteristic 2
- Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?
- 'Randomness' of inverses of $(\mathbb{Z}/p \mathbb{Z})^\times$
- Subgroups of a direct product
- Normal subgroups and cosets
- Sylow $p$-subgroups of finite simple groups of Lie type
- Semisimple ring problem
- Finding the subgroup of $(\mathbb Z_{56},+)$ which is isomorphic with $(\mathbb Z_{14},+)$ by GAP

Suppose $N$ is a normal subgroup of order $84$. By the fact given, $N$ must contain every element of order $3$ and $7$. But $104 > 84$.

Similarly, if $|N| = 56$ and $N$ is normal (of index $3$), $N$ must contain every element of order $2,4$ and $7$. They don’t fit.

Can you continue?

- “If P, then Q; If P, then R; Therefore: If Q, then R.” Fallacy and Transitivity
- Lebesgue measure, Borel sets and Axiom of choice
- Set theory and functions
- Finding values for integral $\iint_A \frac{dxdy}{|x|^p+|y|^q}$ converges
- Convergence types in probability theory : Counterexamples
- Find a bijection from $\mathbb R$ to $\mathbb R-\mathbb N$
- Stochastic processes question – random walk hitting time
- Is there any real number except 1 which is equal to its own irrationality measure?
- Meaning of non-existence of expectation?
- Counting the number of paths on a graph
- What is the exact motivation for the Minkowski metric?
- Can different choices of regulator assign different values to the same divergent series?
- Showing the sequence converges to the square root
- Mapping Irregular Quadrilateral to a Rectangle
- Tangent space in a point and First Ext group