Intereting Posts

Tricky probability problem
Principal Submatrices of a Positive Definite Matrix
What is an example of a nonmetrizable topological space?
perfect square of the form $n^2+an+b$
Is the function $f:\mathbb{R}^2\to\mathbb{R}^2$, where $f(x,y)=(x+y,x)$, one-to-one, onto, both?
Find the smallest number which leaves remainder 1, 2 and 3 when divided by 11, 51 and 91
Where can I learn more about the “else” operation / “else monoid”?
If $f$ and $g$ are integrable, then $h(x,y)=f(x)g(y)$ is integrable with respect to product measure.
n-th derivative of $\sinh^{-1} x$
gcd as positive linear combination
Prove that if $\lim _{x\to \infty } f(x)$,then $\lim_{x\to \infty} f(x)=0$
Sufficient condition for isometry
Is there any diffeomorphism from A to B that $f(A)=B$?
having a question on the symbol $dN_p$ when writing down its correspondence matrix
Existence of a universal cover of a manifold.

I have the group given by the presentation $G= \langle a,b\mid a^2,b^2\rangle$

How can I in general find $G’,G/G’,G”$ ?

thanks for any hints.

- Maximal order of an element in a symmetric group
- Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is an integer $n > 1\mid a^n =a$. Every prime ideal is maximal?
- Example of subgroup of $\mathbb Q$ which is not finitely generated
- Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime?
- Maximal ideal in $K$
- Intersection of two subfields of the Rational Function Field in characteristic $0$

- Equivalence of definitions of prime ideal in commutative ring
- Why $\mathbf Z/(2, 1+\sqrt{-5})\simeq \mathbf Z/(2,x+1,x^2+5)$?
- Every Submodule of a Free Module is Isomorphic to a Direct Sum of Ideals
- Example of modules that are projective but not free; torsion-free but not free
- Norms in $\mathbb{Q}$
- An example of prime ideal $P$ in an integral domain such that $\bigcap_{n=1}^{\infty}P^n$ is not prime
- Hom functors and exactness
- Sum and intersection of submodules
- What does it mean here to describe the structure of this quotient module?
- My proof of $I \otimes N \cong IN$ is clearly wrong, but where have I gone wrong?

In general, $G/G^{\prime}$ is the group attained by making the generators pairwise commute, and is called the *abelianisation* of $G$ (*). As we only have two generators, here $G/G^{\prime}$ is the group $\langle a, b; a^2, b^2, [a, b]\rangle$. I will leave you to work out what this is isomorphic to.

To find the derived subgroup $G^{\prime}$, you can (in general) use something called “Reidemeister-Schreier” (if you want to know more, I once wrote out an example – note that this will only spit out a finite presentation if $G$ is given by a finite presentation and $G/G^{\prime}$ is finite). However, you needn’t do anything quite so fancy here! Indeed, $$G^{\prime}=\langle (ab)^2\rangle.$$ Why? Well, $(ab)^2=abab=a^{-1}b^{-1}ab=[a, b]$, while $\langle (ab)^2\rangle$ is normal in $G$ (why?). Can you see why this is sufficient?

Now, $G^{\prime}$ is cyclic. What does this mean for $G^{\prime\prime}$?

(*) A constructive example of the abelianisation would be something like $\langle a, b, c; aba^{-1}c^{-1}, a^2, b^3\rangle$, then your abelinisation is $\langle a, b, c; aba^{-1}c^{-1}, a^2, c^3, [a, b], [a, c], [b,c]\rangle$…but…as $a$ and $b$ now commute the first relation means that $b=c$, so your group becomes $\langle a, b; a^2, b^3, [a, b]\rangle$. Note that $c^3$ has become $b^3$. Can you see why this is?

Another way to find $G’$ is to notice that $G$ is the free product $\mathbb{Z}_2 \ast \mathbb{Z}_2$. Then $G’$ is the kernel of the canonical projection $\mathbb{Z}_2 \ast \mathbb{Z}_2 \to \mathbb{Z}_2 \times \mathbb{Z}_2$, and

Lemma:Let $H$ and $K$ be two groups. The kernel of $H \ast K \to H \times K$ is free over the set $\{[h,k] \mid h \in H \backslash \{1\}, k \in K \backslash \{1\}\}$.

For example, you can see this answer.

- Construction of cut-off function
- Any tree degree sequence is a caterpillar degree sequence
- Equivalent metrics determine the same topology
- The first-order theory of linear orders given by closed subsets
- Finding the derivative of $x^x$
- Proving that an integral is differentiable
- Is any homomorphism between two isomorphic fields an isomorphism?
- Eisenstein Criterion with a twist
- How can I show that $f$ must be zero if $\int fg$ is always zero?
- Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$.
- If $ f(x) = \frac{\sin^2 x+\sin x-1}{\sin^2 x-\sin x+2},$ then value of $f(x)$ lies in the interval
- how to visualize binomial theorem geometrically?
- Differential equation for Harmonic Motion
- Distribution of Subsets of Primes
- Solve integral(convolution) equation