Intereting Posts

Where is $f(z)=\Re (z)$ differentiable?
Proof of a limit for a recursively-defined sequence
How to solve the following identity?
Uniqueness of a configuration of $7$ points in $\Bbb R^2$ such that, given any $3$, $2$ of them are $1$ unit apart
Cayley graphs on small Dihedral and Cyclic group
How to get determinant of $A$ in terms of tr$(A^k)$?
Numerical Approximation of Differential Equations with Midpoint Method
Group presentations – again
What is wrong with my integral? $\sin^5 x\cos^3 x$
Is there a simple explanation why degree 5 polynomials (and up) are unsolvable?
Let $\sum_{n=1}^\infty \frac{a_n}{3^n}.$ Determine (numerically or not) the limit of the infinite series by choosing $a_n=0$ or $2$ randomly.
This multiple integral notation, has it got a name? $\int dx \int dy \, f(y,x)$
Convergence in weak topology implies convergence in norm topology
Perfect square then it is odd
Help understand the proof of infinitely many primes of the form $4n+3$

**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.

- Find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)$ has order 2
- A nonsplit short exact sequence of abelian groups with $B \cong A \oplus C$
- What is the image of a group homomorphism sending $g$ to $g^p$ for a prime $p$
- In an abelian group, the elements of finite order form a subgroup.
- All subgroups normal $\implies$ abelian group
- Are cyclic groups always abelian?

- Distinguishable painted prisms with six colors (repetition allowed)
- Homogenous polynomial over finite field having only trivial zero
- Proving equivalence relations
- If D is an Integral Domain and has finite characteristic p, prove p is prime.
- $S_n$ acting transitively on $\{1, 2, \dots, n\}$
- Is there a subfield $F$ of $\Bbb R$ such that there is an embedding $F(x) \hookrightarrow F$?
- Galois, normal and separable extensions
- Peano axioms with only sets and mapping
- Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?
- A ring problem in Bhattacharya's book “Basic Abstract Algebra”

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 $G/H\to G$.

- Evaluating $\sum\limits_{n=0}^{20} \frac{(-1)^{n}2^{n+1}}{3^{n}},$
- System of Equations: any solutions at all?
- How to check if the lines are coplanar?
- Interesting math-facts that are visually attractive
- Orthogonal Projection onto the $ {L}_{1} $ Unit Ball
- Why do the even Bernoulli numbers grow so fast?
- What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?
- What can we say about $f$ if $\int_0^1 f(x)p(x)dx=0$ for all polynomials $p$?
- Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E$ an element of the dual space of $C_c^∞(G)$?
- Find limit with factorial
- How many function are there?
- Distance between Unilateral shift and invertible operators.
- The rank and eigenvalues of the operator $T(M) = AM – MA$ on the space of matrices
- About evaluating $\mathcal{L}^{-1}_{s\to x}\bigl\{\frac{F(s)}{s}\bigr\}$ by considering contour integration with different entire functions $F(s)$
- Digit function properties