Intereting Posts

When is the group of units in $\mathbb{Z}_n$ cyclic?
Find $\lim_{n\to\infty}\sqrt{6}^{\ n}\underbrace{\sqrt{3-\sqrt{6+\sqrt{6+\dotsb+\sqrt{6}}}}}_{n\text{ square root signs}}$
$\frac{(10!)!}{(10!)^{9!}}$ is an integer
How to solve this equation for $x$ with XOR involved?
Can one tell based on the moments of the random variable if it is continuos or not
prove that the sphere with a hair in $IR^{3}$ is not locally Euclidean at q. Hence it cannot be a topological manifold.
Are two subgroups that contain a common element conjugate iff they are conjugate under the normalizer?
Prove$ L^2$ inner product satisfies positivity
$a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle
Is my understanding of quotient rings correct?
Use of determinants
Why can't epsilon depend on delta instead?
what is exactly analytic continuation of the product log function
Show that $\lim\limits_{y\downarrow 0} y\mathbb{E}=0$.
Is there a cubic spline interpolation with minimal curvature?

Let $G$ be a group of order $2014=2\times 19\times 53$. The Sylow Theorems give that the Sylow-19 and 53 subgroups are unique and normal. Let these be $H$ and $K$, respectively. Then since $H$ and $K$ intersect trivially, $HK$=$H \times K$=$C_{19} \times C_{53} \cong C_{1007}$ is a (normal) subgroup of $G$. But this means that $G$ is either the direct product with $C_2$, giving $C_{2014}$, or the semidirect product over the only automorphism of order 2, giving $D_{2014}$.

I guess the 2 groups I’m missing are $C_{19} \times D_{106}$ and $C_{53} \times D_{38}$. Is that correct? In either case, what was the flaw in my argument above?

- What are the length of the longest element in a Coexter group for every type?
- Does $\displaystyle \frac{G}{H}$ $\simeq$ $\displaystyle \frac{G}{K}$ $\Rightarrow$ $H$ $\simeq$ $K$?
- Homomorphism and normal subgroups
- semi direct of quaternionic group
- Alternating and special orthogonal groups which are simple
- Understanding the proof of $|ST||S\cap T| = |S||T|$ where $S, T$ are subgroups of a finite group

- Structure Descriptions (GAP) in semigroups
- Subgroups of $S_4$ isomorphic to $S_3$ and $S_2$?
- The center of a non-Abelian group of order 8
- $G$ be a free abelian group of rank $k$ , let $S$ be a subset of $G$ that generates $G$ then is it true that $|S| \ge k$?
- Compatibility of direct product and quotient in group theory
- If $h$ divides $|G|$, not necessary that $G$ has a subgroup of order $h$
- Showing a free abelian group is generated by its basis
- Is $\mathbb{Q}/\mathbb{Z}$ isomorphic to $\mathbb{Q}$?
- Is there a convenient way to show that the symmetric group $S_4$ has a subgroup of order $d$ for each $d|24$?
- If I have the presentation of a group, how can I find the commutator subgroup of it?

Note that

$$\text{Aut} (C_{1007})\cong C_{18}\times C_{52}$$

so $\;C_2\;$ can act as inversion *only* on the generator of $\;C_{19}\;$ , *only* in the generator of $\;C_{53}\;$ or on both. These three non-trivial actions give you three semidirect products…

- How to solve an hyperbolic Angle Side Angle triangle?
- Let $W_1$ and $W_2$ be subspaces of a finite dimensional inner product space space. Prove that $(W_1 \cap W_2)^\perp=W_1^\perp + W_2^\perp $
- Intuitive reasoning why are quintics unsolvable
- Block inverse of symmetric matrices
- Cholesky decomposition and permutation matrix
- How to solve for multiple congruence's what aren't relatively prime.
- Given the inverse of a block matrix – Complete problem
- Connecting square vertexes with minimal road
- Proving limit with $\log(n!)$
- Find all the linear involutions $f: E \to E$, where $E$ is a finite-dimensional real vector space
- Reflection with respect to a parabola
- Definition of “maximal” and “minimal”
- Prove that $e^x\ge x+1$ for all real $x$
- Is a stochastic process being continuous a.e. completely determined by its law?
- Predicate logic for statements about functions?