Intereting Posts

Sample variance converge almost surely
$\dim C(AB)=\dim C(B)-\dim(\operatorname{Null}(A)\cap C(B))$
Theorem 8.17 , Chapter II, Hartshorne
$\sum_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n}?$
Computing the order of $dx$ at the infinity point of an elliptic curve
$\operatorname{Soc}(\operatorname{Aut}( G))$ is isomorphic to $G$, for $G$ a nonabelian, simple group.
Operator: not closable!
Proving that $\lim\limits_{x\to 0}\frac{f(x)}{g(x)}=L$ implies $\lim\limits_{r\to 0}\frac {\int_R f(ry)h(y)\,dy}{\int_R g(ry)h(y)dy}=L$
Reference request for algebraic Peter-Weyl theorem?
Prove that $h'(t)=\int_{a}^{b}\frac{\partial\phi}{\partial t}ds$.
How to prove or disprove statements about sets
Show that a prime divisor to $x^4-x^2+1$ has to satisfy $p=1 \pmod{12}$
Volterra integral equation of second type solve using resolvent kernel
Prove that the greatest integer function: $\mathbb{R} \rightarrow \mathbb{Z}$ is onto but not $1-1$
If $A,B\in M(2,\mathbb{F})$ and $AB=I$, then $BA=I$

Studying for an exam, a review question…

Given $G=\langle x,y|x^4=y^4=e,xyxy^{-1}=e\rangle$.

- Show $|G|\leq16$.

For this, I want to consider that $x^3=x^{-1}$ and $y^3=y^{-1}$ based on our assumptions. I am a little lost as to how to put the second part, $xyxy^{-1}=e$ to show there are no more than 16 elements.

- Seifert-van-Kampen and free product with amalgamation
- let $H\subset G$ with $|G:H|=n$ then $\exists~K\leq H$ with $K\unlhd G$ such that $|G:K|\leq n!$ (Dummit Fooote 4.2.8)
- Prove the _Chinese Remainder Theorem_
- Show that $x^2 + x + 12 = 3y^5$ has no integer solutions.
- A finite commutative ring with the property that every element can be written as product of two elements is unital
- Showing that the direct product does not satisfy the universal property of the direct sum

- If $|G|=16$, find the center of the group and find a group that is isomorphic to $G/\langle y^2\rangle$.

I’m pretty sure the group is centerless.

- Could I see that the tensor product is right-exact using its universal property and the Yoneda lemma?
- An ideal whose radical is maximal is primary
- Is every group of order $21$ cyclic?
- Prove that the group $G$ is abelian if $a^2 b^2 = b^2 a^2$ and $a^3 b^3 = b^3 a^3$
- Principal ideal domains that are not integral domains
- Primes in a Power series ring
- Integral Basis for Cubic Fields
- Definition of a monoid: clarification needed
- Given fields $K\subseteq L$, why does $f,g$ relatively prime in $K$ imply relatively prime in $L$?
- Cardinality of $GL_n(K)$ when $K$ is finite

For the first part note that $yx=x^3y$ from the relations, so any element can be written as $x^ay^b$

For the second part all the forms $x^ay^b$ are distinct – otherwise the number of elements falls below $16$. If you have no better idea show that $x^ay^bx^cy^d=x^{a+3^bc}y^{b+d}$ and identify elements of the centre from this.

That should help you to complete things.

First show that all elements of $G$ can be written as $x^iy^j$ with $0\leq i \leq 3$ and $0\leq j \leq 3$. This tells you that $|G|\leq 16$. Then find a semi-direct product of two cyclic groups of order $4$ that is the image of your group G. This shows that $|G|\geq 16$.

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- Why isn't the Ito integral just the Riemann-Stieltjes integral?
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- Is my intuition of “If $p \mid ab$ then $p \mid a$ or $p \mid b$” correct?
- Diagonalizable matrices with complex values are dense in set of $n\times n$ complex matrices.
- Is my understanding of quotient rings correct?
- Show that the odd prime divisors of the integer $n^2+n+1$ which are different from $3$ are of the form $6k+1$.
- Proof verification for proving $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ by induction
- Is there a definitive guide to speaking mathematics?
- Show that $I$ is an ideal
- Minimum / Maximum and other Advanced Properties of the Covariance of Two Random Variables
- clarification on Taylor's Formula
- If, in a triangle, $\cos(A) + \cos(B) + 2\cos(C) = 2$ prove that the sides of the triangle are in AP
- Does Bezout's lemma work both ways.
- Finding inverse in non-commutative ring