# Tagged Questions

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### Groups of Order 2 with subgroups

Let G be an abelian group and $a,b\in G$ be two distinct elements with a and b or order $2$. Show that $H=\{e,a,b,ab\}$ forms a subgroup and write out its multiplication table. Justify why all the ...
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### $G$ a finite group $n$-abelian goup and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then to show $G$ is abelian [duplicate]

Let $G$ be a finite group and $n$ be a given positive integer such that $(ab)^n=a^nb^n , \forall a,b \in G$ and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then how to prove that $G$ is abelian ? If I can show ...
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### Complement but not direct summand; Lam, Lectures on Modules and Rings, Example 6.17(5)

Let $S$ be a submodule of an $R$-module $M$. A submodule $C⊆M$ is said to be a complement to $S$ (in $M$) if $C$ is maximal with respect to the property that $C∩S=0$. (This does exist by Zorn Lemma.) ...
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### When an Abelian group is cyclic

Let G be a finite abelian group.It contains a non trivial subgroup which is contained in every non trivial subgroup.Then G must be cyclic. This is a problem of Herstein book(Pg 108,#11 2nd edition).I ...
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### From Automorphism to abelian ness … in a finite group

Let $G$ be a finite group such that for any two non-identity elements $a,b$ in $G$ , there is an Automorphism of $G$ sending $a$ to $b$ , then is it true that $G$ is abelian ?
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### Does the following binary operation form a group on a set with 3 elements? (multiple identities?)

Let S = {a, b, c}. *| a b c ----------- a| a b c b| b a a c| c a a This seems to have all the desired characteristics of a group, however, both b and c ...
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### Commutators Calculus

I was trying to understand the above Corollary but I have a problem, namely why in the second to last line $A_0 \leq \zeta_p(G)$? Any ideas? Definitions By recurrence we define $[x,_0\, y]=x$; ...
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### $n$th power map is an automorphism implies abelian group?

If $G$ is a finite group and $\phi(x) = x^n$ is an automorphism of $G$ does this imply $G$ is abelian? I've been reading this page. Def: A group $G$ is said to be $n$-abelian if $(ab)^n=a^nb^n$ ...
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### Counting homomorphisms by the order of their images

I am trying to count homomorphisms from $\mathbb Z^r$ to $(\mathbb Z/m)^n$ while keeping track of the order of the image of each map. In other words, for each integer $k$ dividing $m^n$, I want to ...
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### Simple proof of the structure theorems for finite abelian groups

Many proofs of the structure theorems for finite abelian groups first reduce to the problem to $p$-groups, which is fine and is an important technique. However, it seems to me that a simple proof can ...
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### Orbits under action of a subgroup on the set of conjagtes of a second subgroup

i have the following question: Let $A\leq B\leq G$ be finite groups. Then $G$ acts naturally via conjugation on the set of conjugates $A^G$. It's trivial, that there is only one orbit under this ...
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### If $G$ has more than one nontrivial elements with order 2, how to show that $\prod^n_{x=1}a_x=1$? [duplicate]

Let $G$ be an abelian group of order $n$, and $a_1,a_2,...a_n$ its elements. If $G$ has more than one nontrivial elements with order $2$, how to show that $\prod^n_{x=1}a_x=1$?
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### Prove $G$ is abelian if $f(f(x)) = x$?

Let $G$ be a finite group and $f$ an automorphism such that $f(f(x)) = x$, and $f(x) = x$ if and only if $x=e$. Prove that $G$ is abelian and $f(x) = x^{-1}$. My attempt: ...
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### How to find group homomorphisms from one group to another

I am trying to figure out all the homomorphisms from $\mathbb{Z}_2\times\mathbb{Z}_2$ to $\mathbb{Z}_2$. Is there a good process for doing such a think? I am getting lost...
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### Abelian group generators and relations

(a) Define what it means for an abelian group to be finitely generated. Explain the terms elementary divisors and rank of $G$ and describe the structure theorem for finitely generated abelian ...
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### Rank of $(G/H)/(G/H)_t$ where $G$ is finitely generated abelian and $H$ is a subgroup.

Let $G$ be a finitely generated abelian group and $H$ be a subgroup. Let subscript $t$ denote the torsion subgroup. If $G/G_t$ is free of rank $n$ and $H/H_t$ is free of rank $m$, it is easy to embed ...
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### $\langle x \rangle$ is a direct summand of a finite abelian group where $x$ is maximal order [duplicate]

Let $x$ be an element of a finite abelian group $G$ where $x$ has maximal order. Then I want to show that $\langle x\rangle$ is a direct summand of $G$. Note that I do not want to use finite abelian ...
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### $\psi (m)\leq \phi (m)$ or $\psi (m) \geq \phi (m)$ when $\psi (m)\neq 0$?

(This is different than If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic) I was trying to solve this: Let $G$ be a finite abelian group of order $n$ for which the ...
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### Why is $U(12) = U_{4} (12) ~ U_3(12)?$

Why is $U(12) = U_{4} (2)~ U_3(12)$ Attempt: Any subgroup $U_k(n) = \{x \in U(n)~~|~~x \mod k=1 , k ~|~n \}$ Hence : if $U(12) =\{1,5,7,11\}$ then : $U_4(12) =\{1,5\}$ and $U_3(12)=\{1,7\}$ We see ...
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### Uniqueness FTOFAG

How do you prove uniqueness for the fundamental theorem of finite abelian groups? The book I'm using has this not very well written proof that I can't follow. So following this proof, I multiply by ...
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### Subgroups of a finite elementary abelian group.

I am looking for a method to calculate number all subgroups of a finite elementary abelian $p$-group. Suppose $G$ be an elementary abelian $p$-group of order $p^n$. A proper subgroup $H$ of $G$ is ...
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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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### Evidence about the group $\left ( (\mathbb{Z}/p\mathbb{Z})^*,\underset{p}{ \odot} \right )$

Be $p$ an odd prime number. Show that the group $\left ( (\mathbb{Z}/p\mathbb{Z})^*,\underset{p}{ \odot} \right )$ has a unique element of order $2$, namely $\overline{p-1}$, and show that ...
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### Be $G=\{ e,g_1,g_2,\ldots, g_n \}$, $|G|=n+1$. Suppose $G$ has a unique element of order $2$, say $g_1$. Show that $eg_1g_2\ldots g_n=g_1$.

Be $G=\{ e,g_1,g_2,\ldots, g_n \}$ an abelian group of order $n+1$. Suppose $G$ has a unique element of order $2$, say $g_1$. Show that $eg_1g_2\ldots g_n=g_1$. I have serious difficulties with ...
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### $g\in G$ maximal order in $G$ abelian then $G=\left<g\right>\oplus H$

If $g\in G$ is an element of maximal order in a finite abelian group $G$ then exists $H\leq G$ such that $G=\left<g\right>\oplus H$ Attempt: Using fundamental theorem I know that ...
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### Subgroups of finite Abelian groups

I am interested in finding all of the subgroups (up to isomorphism) of a finite Abelian group $A$. I know the following: -- A finite Abelian group $A$ can be represented as a direct product of ...
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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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### 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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### Using the conjugacy class equation [duplicate]

Let $G$ be a group of order $p^2$. Use the class equation to prove that $G$ is abelian. The conjugacy class equation, at least how I remember it, is  |G| = |Z(G)|+\sum_{x\in I \backslash Z(g)} ...
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### Subgroups of a finite abelian group

Let $G$ be a finite abelian group, and let $K$ be a subgroup of $G$. Does $G$ necessarily have a subgroup $H$ such that $H\cong G/K$ and $H\cap K=\langle 0\rangle$? I'm not sure where to start.
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### Finite abelian p-group and an element of maximal order

I'm studying for an exam and I'm having trouble understanding the proof given for the following statement: Suppose $G$ is a finite abelian $p$-group and $a \in G$ has maximum order, then there ...
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### Question about Finite Abelian Groups [duplicate]

Let $(G, .)$ be a finite abelian group, $G=\{x_1, ..., x_n\}$ and let $x=x_1. \cdots. x_n$. Show that $x^2=e$ Suppose $G$ has no element of order 2 or that $G$ has more than one element of order 2. ...
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### an order of automorphism group of finite abelian group

This is problem of Rotman's Exercise 7.9(i). If $G$ is finite abelian group with $|G| >2$, then $\operatorname{Aut}(G)$ has even order. How can I approach to this problem? Could you suggest ...
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### Working with special cases of the converse of Lagrange's theorem

I am to answer true/false to statements on the form: Every abelian group of order divisible by $n$ contains a cyclic subgroup of order $n$. This follows directly from Cauchy's theorem when $n$ ...
Let $Z$ be a finite group and denote $k(Z)$ the sum of the orders of all the elements of the group $Z$. I have to determine min $k(Z)$ and max $k(Z)$ when $G$ goes through the set of the abelians ...
I'm trying to prove the following: Let $G$ be an abelian group of order 72. Show that $G$ has exactly one subgroup of order 8. I know by theorem that $G$ must have at least one subgroup of order ...