Intereting Posts

For which topological spaces $X$ can one write $X \approx Y \times Y$? Is $Y$ unique?
Example of topological spaces where sequential continuity does not imply continuity
When the quadratic formula has square root of zero, how to proceed?
What is “Approximation Theory”?
What does Universal mapping property for a free monoid mean?
Smallest graph with automorphism group the quaternion $8$-group, $Q_8$
The prime numbers do not satisfies Benford's law
Cover and extension of a Lie group
Contraction mapping in an incomplete metric space
About frullani integral
21 sided regular polygon and its diagonals
Where to begin in approaching Stochastic Calculus?
Counting the Number of Integral Solutions to $x^2+dy^2 = n$
Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.
Group actions and associated bundles

**STATEMENT**: Let $H$ be a subgroup of a finite abelian group $G$. Show that $G$ has a subgroups that is isomorphic to $G/H$.

**QUESTION**: Could someone offer a proof using dual groups. I have found one use the fundamental theorem of finitely generated abelian groups, but I cannot seem to find on using dual groups.

- number of subgroups index p equals number of subgroups order p
- For abelian groups: does knowing $\text{Hom}(X,Z)$ for all $Z$ suffice to determine $X$?
- Any finitely generated subgroup of $(\mathbb{Q},+)$ is cyclic.
- Shorter proof for some equvalences
- Does $A\oplus \mathbb{Z}\cong B\oplus \mathbb{Z}$ imply $A\cong B$?
- Automorphisms of abelian groups and Choice

- Algebraic operations with tensor products and direct sums.
- Prove that $x-1$ is a factor of $x^n-1$
- On Symmetric Group $S_n$ and Isomorphism
- Help me prove equivalently of regular semigroup and group.
- Proof that no permutation can be expressed both as the product of an even number of transpositions and as a product of an odd number of transpositions
- Finitely generated group which is not finitely presented
- Normal subgroup question/example
- Multiplicative group of integers modulo n definition issues
- Can every ideal have a minimal generating set?
- Groups of order $pqr$ and their normal subgroups

Let $S^1$ be the circle group. Consider the exact sequence:

$$0\to H\to G \to G/H \to 0$$

Applying the functor $\operatorname{Hom}(-,S^1)$, we get another exact sequence:

$$0\to \operatorname{Hom}(G/H,S^1) \to \operatorname{Hom}(G,S^1) \to \operatorname{Hom}(H,S^1)$$

Now, it so happens that $\operatorname{Hom}(G,S^1) \cong G$ for a finite abelian group $G$—though I don’t know if there’s a good way to prove that without using the fundamental theorem. But if you accept it, then this sequence gives us a (non-canonical) injection $G/H\to G$.

- Expressing the integral $\int_{0}^{1}\frac{\mathrm{d}x}{\sqrt{\left(1-x^3\right)\left(1-a^6x^3\right)}}$ in terms of elliptic integrals
- How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?
- Hilbert dual space (inequality and reflexivity)
- linear algebra books with many examples
- Proving that all integers are even or odd
- Differentiable function, not constant, $f(x+y)=f(x)f(y)$, $f'(0)=2$
- Taylor series expansion of $\log$ about $z=1$ (different branches)
- Expected number of people sitting in the right seats.
- $C()$ is not weakly sequentially complete.
- Bounds on off-diagonal entries of a correlation matrix
- How to figure of the Laplace transform for $\log x$?
- Homology and Graph Theory
- Wire mass line integral
- How to test whether two group presentations are isomorphic
- In how many ways can we arrange 4 letters of the word “ENGINE”?