# Tagged Questions

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### Prove that $U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$

If $m = n_1 n_2 \cdots n_k$ where $\gcd(n_i~,n_j)=1 ~~ \forall i \neq j$, then prove that: $$U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$$ where $\times$ refers to the ...
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### Prove that a(mn)=a(m)a(n), (n,m)=1

Given a positive integer $n$ where $a(n)$ is the number of non-isomorphic abelian groups of order n. 1) Prove that $a(mn)=a(m)a(n), (n,m)=1$ 2) Prove that $a(p^k)$ is the number of partitions of k, ...
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### if $G' <H < G$ then $H$ is normal in $G$.

if $G' <H < G$ then $H$ is normal in $G$. ($G'$ is the commutator subgroup of $G$.) This is what I do: because $G' < H$ we have $\frac{H}{G'} \triangleleft \frac{G}{G'}$. because ...
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### How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...
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### Abelian and residually finite groups

If $G$ is a finitely generated abelian group then $G$ is residually finite. I don't know if the result holds or not. I tried to follow the definition but could not go far. Any hint will be highly ...
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### Maximal $\mathbb{Z}$-submodules in $\mathbb{Q}$

Is it true that $\mathbb{Q}$ viewed as $\mathbb{Z}$-module ( i.e. abelian group ) has not maximal $\mathbb{Z}$-submodules ? Why ?
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### Order of elements in a disjoint cycle

What's the difference between a subgroup and a cyclic subgroup? $A_4 = \{e,(123),(132),(124),(142),(134),(143),(234),(243),(12)(34),(13)(24),(14)(23)\}$ And if I was looking for a subgroup of order ...
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### Order of elements in a disjoint cycle

Just wondering how to find the order of each element in this group: $A_4 = \{e,(123),(132),(124),(142),(134),(143),(234),(243),(12)(34),(13)(24),(14)(23)\}$ I tried writing each elements not in ...
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### Let $A$ finitely generated abelian group, and $A_1 \le A$. I have to prove that $rk(A)=rk(A_1)+ rk(A/A_1)$

Let be $A$ a finitely generated (f.g.) abelian group, and let be $T$ its torsion, then by the structure theorem of f.g. abelian gruops we have that $A/T \simeq \mathbb{Z}^d$, so we can define $b$ the ...
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### uniqueness of groups in an exact sequence

I was wondering how unique are the groups making up to an exact sequence. Suppose we have three groups $A, B, C$ such that the sequence $$A \rightarrow B \rightarrow C$$ is exact. I wanted to know ...
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### If $G$ is a finite union of some of its abelian subgroups, then the index of the center of the group is finite

If $G$ is a finite union of some of its abelian subgroups, then the index of the center of the group is finite Would I not simply state that by Lagrange's theorem, $Z(G)$ can divide into the ...
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### Shorter proof for some equvalences

Let $(G,\cdot)$ be a group show that A) $G$ is abelian B) For all $x,y\in G: (xy)^{-1}=x^{-1}y^{-1}$ C) For all $x,y\in G: (xy)^{2}=x^{2}y^{2}$ D) There existst an $n\in \mathbb{Z}$ such that for ...
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### Is the group $(G,*)$ abelian?

Let $(G,*)$ be a finite group in which the sets $C_a$={$x\epsilon G$|$ax=xa$} have the same cardinality, for all $a \epsilon G$ \{e}. My question is: is the group abelian?
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### Using the Fundamental Theorem of Finite Abelian Groups

Let $G$ be a finite abelian group and let $p$ be a prime that divides the order of $G$. Use the Fundamental Theorem of Finite Abelian Groups to show that $G$ contains an element of order $p$. The ...
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### Explicitly computing the isomorphism class of the tensor product of two finite abelian groups

How do I compute the isomorphism class of $A\otimes_\mathbb{Z} B$, where $A$ and $B$ are abelian of finite order? I can do this for a few examples, but I am unsure of how to proceed in the ...
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### Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...
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