Group of order $p^nq^m$ February 13

Let $G$ be a semi-direct product of a finite $p$-group and a finite $q$-group where $p$ and $q$ are prime numbers. We know that $G$ is soluble and every normal minimal of $G$ is elementary abelian. If $G$ does not contain a normal minimal subgroup of

Solvable group of order $p^nq^m$ February 13

Let $G$ be a semi-direct product of a $p$-group and a $q$-group where $p$ and $q$ are prime number. If $G$ does not contain a normal minimal subgroup of order $q$ what we can say about $q$-sylow of $G$? For example is it cyclic?

Rotating Solids and Integrals February 13    3

So we are rotating solids. I was given the following information:$$R=\left\{(x,y) : 1 \le y \le \sqrt{4-x^2}, 1 \le x \le 2\right\}$$I am supposed to find the volume of the solid when we rotate $R$ around the $y$-axis. I drew the graph and I'm pretty

Integral $\frac{sin(x)}{x}$ finite domain February 13

I have seen a question asking to find the value of $\int_{-100}^{100} \frac{\sin{x}}{x} dx$.I have to confess that I didn't think this was possible. If I expand the $\sin$ using Taylor series, then unless the endpoints of the domain lie inside $(-1,1

Independence Number Proof Explanation February 13

Independence Number Proof Explanation
In the following proof it states that "$v_i$ is less than or equal to the independence number for all $i$." Why is this true? I know what an independence number represents, I am st ...

Villarceau circle as a Loxodrome February 13

Villarceau circle as a Loxodrome
A circular Clifford torus (radius at flat circle = h, section radius $ a , a<h $ ) is cut by a plane at an angle $ \cos \alpha = a/h \tag{1} $ centrally to the symmetry axis, the line of ...

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