Intereting Posts

Wikipedia Proof of Skolem-Noether Theorem
Local homeomorphisms which are not covering map?
Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$
What's wrong with this argument? (Limits)
Definition of the algebraic intersection number of oriented closed curves.
A maximal ideal among those avoiding a multiplicative set is prime
Please verify my proof of: There is no integer $\geq2$ sum of squares of whose digits equal the integer itself.
About a measurable function in $\mathbb{R}$
Prove by contradiction or contrapositive? If $|x+y|<|x|+|y|$, then $x<0$ or $y<0$.
How could it possible to factorise $x^8-1$ in product of irreducibles in the rings $(\mathbb{Z}/2\mathbb{Z})$ and $(\mathbb{Z}/3\mathbb{Z})$?
how to solve linear equations involving modulo?
How do I show that two groups are not isomorphic?
Ordered pairs in a power set
Are the $\mathcal{C}^k$ functions dense in either $\mathcal{L}^2$ or $\mathcal{L}^1$?
Is metric (Cauchy) completeness “outside the realm” of first order logic?

From Dummit & Foote, as usual, $\S$ 2.4 #14.

A group $H$ is called

finitely generatedif there is a finite set $A$ such that $H = \left \langle A \right \rangle$

(a) Prove that every finite group is finitely generated.

(b) *(Prove that $\mathbb{Z}$ is finitely generated* – I am comfortable with my proof of this, *viz* $\left \langle 1, -1 \right \rangle = \mathbb{Z}$ )

(c) Prove that every finitely generated subgroup of the additive group $\mathbb{Q}$ is cyclic [If $H$ is a finitely genereated subgroup of $\mathbb{Q}$, show that $H \leq \left \langle \frac{1}{k} \right \rangle$, where $k$ is the product of all the denominators which appear in a set of generators for $H$

- Sufficient conditions for being a PID
- Why is quadratic integer ring defined in that way?
- Infinite Degree Algebraic Field Extensions
- Reference request: algebraic methods in geometry
- Order of products of elements in a finite Abelian group
- Let $R$ be a ring such that for all $a,b$ in $R$, $(a^2-a)b=b(a^2-a)$. Then $R$ is commutative

Logically, I’m having a hard time getting started on both (a) and (c). For (a), I **knew** that $H$ generates itself (if that’s the correct way to say it), i.e. $\left \langle G \right \rangle = G$ **before** looking at this relevant wikipedia page, but can’t seem to articulate this in a manner consistent with the (currently inaccessible) definition of

$\left \langle A \right \rangle = \bigcap_{A\subseteq H; H \leq G} H$

or the proven result that this subgroup is the set of all products (… a bunch of elements in $G$ with exponents…

“words”, they call them)

Does this just follow simply from the above definition?

(c) I know (think?) that to show $H$ cyclic, I must take $h \in H$ and show that $h = a^k$, for some $k \leq |H|$ … or something like that. But then there’s that *hint*. Where do I begin?

Thanks for your help.

- $GL(2,\mathbb{Z})$ and nilpotency
- What are all the intermediate fields of $\mathbb{Q}\big(\sqrt{3+\sqrt{5}}\big)$ containing $\mathbb{Q}$?
- Examples of non-isomorphic fields with isomorphic group of units and additive group structure
- Isomorphic subfields of $\mathbb C$
- Using orbit stabilizer theorem
- Ring of $p$-adic integers $\mathbb Z_p$
- Topology induced by the completion of a topological group
- Number of elements in the quotient ring $\mathbb{Z}/(X^2-3, 2X+4)$
- Understanding quotients of $\mathbb{Q}$
- Embedding $S_n$ into $A_{n+2}$

For part (a), yes it follows from the definition. Since $G \subseteq G$ and $G \leq G$, we see that $G$ is one of the groups in the intersection, so $\langle G \rangle = \bigcap_{A\subseteq H; H \leq G} H \subseteq G$. But we have that $G \subseteq \langle G \rangle$, also by definition, so $\langle G \rangle = G$.

Hint for part (c): Find the greatest common divisor (over $\mathbb{Q}$) of the finitely many generators.

Addressing the other (equivalent) definition of $\langle G\rangle$:

Every element $g$ of $G$ is a product of elements of $G$, namely the product of length one with the single factor $g$.

Hence $G\subseteq\langle G\rangle$.

Since $G$ is closed under multiplication and taking inverses, $\langle G\rangle\subseteq G$. It follows that $G=\langle G\rangle$.

Additional hint for part (c): Once you have found the $k$ with $H\subseteq\langle\frac{1}{k}\rangle$, recall that every subgroup of a cyclic group is again cyclic.

- How to find $\lim_{x\to 0} \frac{1-\cos x \sqrt{\cos 2x}}{x^2}$
- Confusion regarding Russell's paradox
- Prove that $\exp(x)>0$ using only formal definition of exp
- Intuitive bernoulli numbers
- Forecast equivalence between two steady-state Kalman filters
- Linear algebra on Fibonacci number
- Alternative characterization of $\exp(x), \log x$
- Understanding positive definite kernel
- Is totally disconnected space, Hausdorff?
- Tangent bundle of $S^1$ is diffeomorphic to the cylinder $S^1\times\Bbb{R}$
- What are examples of unexpected algebraic numbers of high degree occured in some math problems?
- How to extend this extension of tetration?
- Is the Cantor set made of interval endpoints?
- Continuous function on metric space
- Representing the tensor product of two algebras as bounded operators on a Hilbert space.