# Tagged Questions

Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$

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### Intervals in divisible ordered groups

Is it true that if $(G,+,0,<)$ is a divisible ordered abelian group with at least two elements, then for $a,b >0 \in G$, there is an injective order preserving map from $[0;a)$ to $[0;b)$? It ...
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### Characters of a Finite Abelian group

Let $C_m$ denote the cylic group of order $m$. Recall $\widehat{C}_m=\{ \psi_j: j\in C_m\},$ where $\psi_j$ is defined by \begin{align} \psi_j(k)=e\Big( \frac{2\pi \sqrt{-1}jk}{m}\Big). \end{align} ...
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### The central product of two cyclic subgroups of prime power order for one $p$ is isomorphic to direct product of two cyclic groups of prime power order

For two groups $G_1, G_2$ and two central subgroups $U_1 \le Z(G_1), U_2 \le Z(G_2)$ which are isomorphic by some given $\mu : U_1 \to U_2$ the central product is the group $$(G_1 \times G_2) / D$$ ...
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### Is there a known formula for the “cyclicity” of a positive integer?

Given a positive integer $n$, let us define that the cyclicity of $n$ is the number of multitsets of cyclic numbers (distinct from $1$) whose product is $n$. For example, the cyclicity of $15$ is $2$, ...
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### if $ab = ba$ for all $a \in X$ and for all $b \in X$ then $\langle X \rangle$ is abelian subgroup of $G$

if $X \subseteq G$ such that $\forall a,b \in X$ we have $ab = ba$ then we should prove that $\langle X \rangle$ is an abelian subgroup of G. its abviouse that $\langle X \rangle$ is subgroup of $G$. ...
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### Is every subgroup of a free abelian group a direct summand?

My guess is NO, because take $G=\mathbb{Z}$ and $F=2\mathbb{Z}$ is a subgroup but not a direct summand.
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### Example of an endomorphism on an abelian group that is not left multiplication

It is well-known that all endomorphisms on the abelian group ($\Bbb{Z}$,+) can be seen as a left multiplication by some element in some ring structure on ($\Bbb{Z}$,+); namely left multiplication by ...
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### Is there a non abelian group of order 759? [closed]

I tried to use Sylow theorems to prove that there is not, but it is not trivial.
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### Number of elements of order $2$ in Abelian groups of order $2^{n}$

I'm self studying some group theory and one of the exercises I came across: Question: Prove that an Abelian group of order $2^{n}, n \in \mathbb{N}$ must have an odd number of elements of order 2. ...
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### Find the number of elements of order 2 and number of subgroups of index 2.

Let $A= \mathbb Z_{60} \oplus \mathbb Z_{45} \oplus \mathbb Z_{12} \oplus \mathbb Z_{36}$. 1) what is the number of elements in A with order 2. 2) what is the number of subgroups of A with ...
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### How to recognize a finitely generated abelian group as a product of cyclic groups.

Let $G$ be the quotient group $G=\mathbb{Z}^5/N$, where $N$ is generated by $(6,0,-3,0,3)$ and $(0,0,8,4,2)$. Recognize $G$ as a product of cyclic groups. Honestly, I do not know how to solve these ...
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### Showing that every subgroup of an abelian group is normal [duplicate]

I'm working on a proof to show that every subgroup of an abelian group is also a normal subgroup. Let $G$ be an abelian group and $H$ an arbitrary subgroup of $G$. I want to show that $gHg^{-1} = H$, ...
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### $G$ is a finite abelian group. For every prime $p$ that divides $|G|$, there is a unique subgroup of order $p$.

$G$ is a finite abelian group. Assume that for every prime $p$ that divides $|G|$, there is a unique subgroup of order $p$. I'd like to prove that $G$ is cyclic. I'm thinking about the approach of ...
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### A finite group is solvable iff the simple factors in a decomposition sequence are abelian

Show that a finite group $G$ is solvable group (in the sense there exists an $n$ such that $G^{(n)}=1$) if and only the simples factors in a decomposition sequence of $G$ are all abelians. I'm not ...
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### Every group with 5 elements is an abelian group

I have tried to prove that every group with 5 elements is an abelian group using following approach. Is this correct: Note: I do do not want to use Lagranges theorem and I do not know why groups with ...
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### Detail in the proof $\text{rank}(G)=\dim_{\mathbb{Q}}(G\otimes\mathbb{Q})$

Let $G$ be a finitely generated $\mathbb{Z}$-module. I want to show that $\text{rank}(G)=\dim_{\mathbb{Q}}(G\otimes\mathbb{Q})$. I've shown that $G\otimes\mathbb{Q}\cong \mathbb{Q}^{\text{rank}(G)}$ ...
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### Proof containing abelian groups.

Let $G \leq S_{999}$ be an abelian subgroup of order $|G| = 1111$. Prove that there exists $i \in$ {$1,2,...,999$} such that $\forall α \in G, α(i) = i$. Okay so I came across this problem and even ...
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### Relations on groups.

Let $G$ be a group and $H$ a subgroup of $G$. Define a relation on elements of $G$ by saying that $a \sim b$ if $b^{-1} a \in H$. This relation is : a) reflexive and symmetric, but transitive only ...
### Create a base that contains $x$ in a finite generated Abelian group
Let $A$ be a finite generated free Abelian group and $x \in A$ such that $\forall y \in A$ $\forall n>1: x\neq ny$. Prove that there is a base generating $A$ that contains $x$. It seems simple ...