# Tagged Questions

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### Rational group algebras and maximal orders

Let $G$ be a finite group, and $\mathbb{Q}[G]$ be the rational group algebra. Then the group ring $\mathbb{Z}[G]$ is an order in $\mathbb{Q}[G]$, but is not in general a maximal order. What can we ...
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### If the automorphism group of a group is cyclic, then the group is commutative [duplicate]

Let $G$ be a group and the $Aut(G)$ group is cyclic $\Rightarrow$ the group $G$ is commutative. I looked at the homomorphism $\varphi : G \rightarrow Aut(G) \ g \mapsto (x \mapsto gxg^{-1})$. Let ...
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### $G$ is an abelian group of order a product of distinct primes $\implies G$ is cyclic?

If $G$ is an abelian group of order $p_1p_2...p_k$ , where $p_1,p_2,...,p_k$ are distinct primes , then is it true that $G$ is cyclic ?
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### Free groups vs. free abelian groups

I'm trying to solve this question in page 74 of Hungerford's book: A free abelian group is a free group (Section I.9) if and only if it is cyclic. I have no idea how to proceed, a solution or a ...
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### Cyclic Factor group abelian proof [duplicate]

Show that if G is nonabelian, then the factor group G/Z(G) is not cyclic. I started to prove this via contrapositive. If G/Z(G) is cyclic, then G is abelian. I'm messing around with elements and ...
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### Number of elements in a group

The group $G$ consists of the binary strings of length $5$ under addition $\mod 2$ in each component. (It is isomorphic to $(\mathbb Z_2)^5$, the direct product of $5$ copies of $\mathbb Z_2$.) Let ...
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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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### Does this condition gaurantee the cyclicity of a finite abelian group?

Let $G$ be a finite abelian group in which there are at most $n$ solutions of the equation $x^n = e$ for each posivite integer $n$. How to determine if $G$ is cyclic or not?
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### Which elements of this cyclic group would generate it?

Let $n$ be a given arbitrary positive integer, and let $U_n$ denote the group of all the positive integers less than $n$ and relatively prime to $n$ under multiplication mod $n$. Then for which values ...
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### When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does $G$ is abelian or $G$ is cyclic implies $\text{Aut }G$ is cyclic?
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### If $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$.

Let $G$ be finite abelian group and $\hat G$ be its character group. I need hint proving that if $a\in G$ and $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$ (the identity element). I can prove it ...
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### Let $G = \{1, a, b, c\}$ be a group of order 4…Exist two groups of order $4$.

Let $G = \{1, a, b, c\}$ be a group of order 4. Show that, if $G$ is cyclic $G \cong \mathbb Z_4,$ and if $G$ is not cyclic then $G \cong K_4.$ It now follows that there are only 2 groups of order ...
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### $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
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### Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic.

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic. You may only assume the Feit-Thompson theorem here and prove in the following ...
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### How do we prove this fact about cyclic groups?

Prove that an Abelian group of order 33 is cyclic. Can we take an element a of order 3 and an element b of order 11 and say, |ab|=33?
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### Prove or disprove that there is an abelian, noncyclic group of order 52.

So I've heard one must invoke Sylow's theorems in order to break down something like this. So far I know that there is a subgroup of order 13 in G, and that it's the only subgroup of order 13 in G. To ...
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### Showing that $U(2^n)$ is not a cyclic group for $n \geq 3$ [duplicate]

Could anyone please explain to me why $U(2^n)$ is not a cyclic group for $n \geq 3$? I need help on this because I have an algebra exam tomorrow. Thanks!
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### $G$ has exactly three subgroups

My attempt for the first: (I would like to get it verified because I didn't use property of a cyclic group) $|G|<\infty$ (since for otherwise $(a^2),(a^3)$ are distinct improper nontrivial ...
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### Finding a subgroup in the Center with order 91

Question: Let G be a group of order $455=5\cdot 7\cdot 13$. Show that exists a normal subgroup $H<G: |H|=91$ and $H\subseteq Z(G)$. Show that G is an Abelian and cyclic group. Solution: So ...
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### Groups - Prove that if $G/Z(G)$ is cyclic then $G$ is abelian [duplicate]

Prove that if $G/Z(G)$ is cyclic then $G$ is abelian. Using this fact and $G$ is a nontrivial group of prime power order, deduce that a group of order $p^2$ , $p$ prime, is abelian.
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### Abelian group admitting a surjective homomorphism onto an infinite cyclic group

I am working on the following problem: Let $G$ an Abelian group and $f: G \to \Bbb Z$ a surjective homomorphism. Prove that $G \cong \ker(f) \times \Bbb Z$ By means of the First Isomorphism ...
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### Determining whether two groups are isomorphic

I am reading "A First Course in Algebra", and there, I am trying to solve the exercises, but there is something i don't understand. How do we understand whether two groups are isomorphic or not? For ...
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### What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$

What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$ and how to find it? I came across the above problem and do not know how to get it? Can someone point me ...
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### How to find subgroups of $\;\;\Bbb Z_2\times \Bbb Z_6$

I am reading a first course in algebra and there is an example saying that "find all the subgroups of $\Bbb{Z}_2\times\Bbb{Z}_6$ and decide which of them are cyclic. I know that ...
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### Understanding a Theorem regarding Order of elements in a cyclic group

This is part of practice midterm that I have been given (our prof doesn't post any solutions to it) I'd like to know whats right before I write the midterm on Monday this was actually a 4 part ...
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### Show that all abelian groups of order 21 and 35 are cyclic.

Show that all abelian groups of order 21 and 35 are cyclic. I have no idea on how to start. Can anyone give some hints?
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### Is $\mathbb{Z}^2$ cyclic?

Is $\mathbb{Z}^2$ cyclic? What does it mean for a group to be cyclic? Is it just that it has one generator? Thanks
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### $G$ finite group, $H \trianglelefteq G$, $\vert H \vert = p$ prime, show $G = HC_G(a)$ $a \in H$

Let $G$ be a finite group. $H \trianglelefteq G$ with $\vert H \vert = p$ the smallest prime dividing $\vert G \vert$. Show $G = HC_G(a)$ with $e \neq a \in H$. $C_G(a)$ is the Centralizer of $a$ in ...
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### Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.

I've tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.
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### Group extensions of cyclic groups

Let $A$ be an infinite cyclic group and $B$ be a cyclic group of order $n$. Suppose $$0 \to A \to G \to B \to 0$$ is a short exact sequence of abelian groups. What could $G$ be? It is clear enough ...
More precisely, I am given a function $f:G\to H$ with the promise that it is a homomorphism. Is there an easy way to determine which homomorphism it is without looking through all of its values? For ...
If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is? Thanks in advance!