Intereting Posts

Transvection matrices generate $SL_n(\mathbb{R})$
When plotting a bell curve from an array of values, is it possible that +/- 2 standard deviations from the mean can fall outside the range of values?
Is there a non-trivial example of a $\mathbb Q-$endomorphism of $\mathbb R$?
Does there exist a Noetherian domain (which is not a field ) whose field of fractions is ( isomorphic with ) $\mathbb C$ ?
Expected Value of Flips Until HT Consecutively
Probability of rolling three dice without getting a 6
Examples of pairewise independent but not independent continuous random variables
What's an easy way of proving a subgroup is normal?
Can somebody explain the plate trick to me?
Cardinality of a $\mathbb Q$ basis for $\mathbb C$, assuming the continuum hypothesis
What is the value of $\sum_{p\le x} 1/p^2$?
How do I prove $\leq $?
Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z $
What book are good to learn about limits and continuity for the first time?
Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$

I am trying to prove the following statement:

Let $G$ be a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K =

\{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times

K$.

From this, we get that $HK \cong H \times K$. Naturally, it would be ideal if I could prove that $HK \cong G$. One issue that I am facing here is the meaning of $G = \langle H, K \rangle$. How is this defined (I mean, more explicitly)? Is my strategy correct or

- Are these two definitions of a semimodule basis equivalent?
- Symmetric Difference Identity
- Does a finite commutative ring necessarily have a unity?
- Field extensions and monomorphism
- Identically zero multivariate polynomial function
- The number of elements which are squares in a finite field.

- Inverse of open affine subscheme is affine
- Is the given group $H$ is a normal subgroup of $S_4$?
- $\mathbb{Q}(\sqrt2,\sqrt3,\sqrt{(2+\sqrt{2})(3+\sqrt{3})})$ is Galois over $\mathbb{Q}$
- Equivalence of tensor reps & tensor products of reps
- Prove that $G = \langle x,y\ |\ x^2=y^2 \rangle $ is torsion free.
- Should every group be a monoid, or should no group be a monoid?
- Irreducibility of $x^5 -x -1$ by reduction mod 5
- Prove $4+\sqrt{5}$ is prime in $\mathbb{Z}$
- What is the image of a group homomorphism sending $g$ to $g^p$ for a prime $p$
- Ideal in a matrix ring

If both $H$ and $K$ are normal in $G$, note that

$$\color{green}{\underbrace{\color{black}{h^{-1}k^{-1}h}}}\color{green}{\underbrace{\color{black}{k}}}\in \color{green}{K} \hspace{15pt}\text{and}\hspace{15pt}\color{blue}{\underbrace{\color{black}{h^{-1}}}}\color{blue}{\underbrace{\color{black}{k^{-1}hk}}}\in \color{blue}{H}$$

so we have $h^{-1}k^{-1}hk\in H\cap K=1$. Thus $h^{-1}k^{-1}hk=1$, so by left multiplying by $kh$ we get $hk=kh$. Since our choice was arbitrary, we must have that $H$ and $K$ commute elementwise.

$\langle H , K \rangle$ is the group generated by all words with letters in $H$ and $K$. Because $H$ and $K$ commute, however, we observe that given any such word, we can rearrange so that all $H$ are on the left and all $K$ are on the right. In particular, every element of $\langle H , K \rangle$ may be *uniquely* expressed as a product $hk$ for some $H\in H$ and some $k\in K$.

Now let $\varphi:\langle H , K \rangle\rightarrow H\times K$. Proving the homomorphic property is the content of this statement. Then, because elements of $G$ are uniquely represented as products $HK$, the isomorphism has a trivial kernel, so it’s injective, so it’s surjective, and we’re finished.

The symbol $\langle H, K \rangle$ is used to denote subgroup generated by elements of $H$ and elements of $K$. Since both $H$ and $K$ are normal subgroups, $HK$ is a subgroup. It should now be straightforward to prove that $HK = \langle H, K \rangle$

- Maths brain teaser. Fifty minutes ago it was four times as many minutes past three o'clock
- Showing a set is a subset of another set
- Vector path length of a hypotenuse
- The Cantor set $K$ in $\mathbb{R}$ homeomorphic to $K\times K$ in $\mathbb{R}^{2}$
- Induced map of homology groups of torus
- PDF of $f(x)=1/\sin(x)$?
- Conditions for integrability
- Rellich's theorem for Sobolev space on the torus
- What is the Definition of Linear Algebra?
- Classifying the factor group $(\mathbb{Z} \times \mathbb{Z})/\langle (2, 2) \rangle$
- inverse Laplace transfor by using maple or matlab
- Are these two statements equivalent?
- Proof of the affine property of normal distribution for a landscape matrix
- Derivation of the general forms of partial fractions
- $K\subseteq \mathbb{R}^n$ is a compact space iff every continuous function in $K$ is bounded.