Are all finite groups isomorphic to the multiplicative group of a ring. If I know the multiplicative group, can I define the addition as a function of 2 variables for all groups?
68270
Motivated by :   Which finite groups are the group of units of some ring? Which finite groups are the group of units of some finite ring? Which finite abelian groups are the group of units of some commutative ring? Which finite abelian groups are the group of units of some finite commutative ring?   It  that even for fields this is not simple to answer.  I expect the general question (Which groups are the group of units of some ring?, with no finiteness hypotheses) is even harder.
55480
Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not.  pf) Suppose that these are isomorphic. Note that $\mathbb{Z}\times \mathbb{Z}$ is a subgroup of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times\left \{ 0 \right \}$ is a subgroup of $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$. Since $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times\left \{ 0 \right \}$ are isomorphic, $\mathbb{Z}\times \mathbb{Z}/\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}/\mathbb{Z}\times \mathbb{Z}\times \left \{ 0 \right \}$ are isomorphic. But the first one is isomorphic to the trivial group and the second one is isomorphic to $\mathbb{Z}$. It is a contradiction.  Is my proof right? If not, is there another proof?
73658
I am looking for a correct proof of this statement: If $G$ is a group such that $G/Z(G)$ is cyclic, then $G$ is commutative.  Proof: $G/Z(G)$ is isomorphic to $\operatorname{Inn}(G)$ and is cyclic, and then for every $a$ and $b$ in $G$ the inner isomorphisms $\gamma_a$ and $\gamma_b$ satisfy $\gamma_a \gamma_b = \gamma_{ab} = \gamma_{ba} = \gamma_b \gamma_a$, and therefore for every $a,b \in G$, $ab = ba$.  Is that proof complete, or am I missing something? Thanks a lot for the help.
38583
Let $G$ be a group and $H_1$ and $H_2$ be two subgroups of $G$. Is it true that if $H_1$ is isomorphic to $H_2$ then $G/H_1$ is isomorphic to $G/H_2$? If that is not true then in what conditions that holds? Any help will be useful.. Thanks
35154
Given two isomorphic groups G and H, is it possible that two or more functions define their isomorphism? Also, is it possible that another group say, L is isomorphic to G but not to H?
99751
Prove that the additive group $ℚ$ is not isomorphic with the multiplicative group $ℚ^*$. Prove that $ℚ^*_{&gt;0}$ is not isomorphic with $ℚ$.
101981
Here I am under the impression that $2\mathbb Z$ and $3\mathbb Z$ are the sets of even numbers and multiples of $3$ respectively and the operations are usual addition and multiplication. This is an exercise in Fraleigh Abstract Algebra.  I believe that $2\mathbb Z$ and $3\mathbb Z$ are isomorphic as groups on addition.  I started a proof by supposing that there was a ring isomorphism, $f$,  in hopes of finding a contradiction with $f(x\cdot y) = f(x)\cdot f(y)$.
24475
Show that up to isomorphism there exist only two groups of order 10: $C_{10}$ and $D_{10}$. I need some help on this question. I only know the basic definitions of isomorphism. Any hints?
23752
The question (from my exam) quoted verbatim:  Show that the additive group of $\mathbb{Z}[x]$ is isomorphic to the group of positive rational numbers under multiplication.  First of all, is this $\mathbb{Z}[x]$ the same as $\mathbb{Z}_x$ (addition modulo $x$)? Secondly, how do I still get the isomorphism? Thanks
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Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field.  Proof.  Surpose that: $\phi : (A, +) \rightarrow  \mathbb{Z} $ is an isomorphism. Then there is some $r∈A $ such that $ϕ(r)=1_\mathbb{Z}$. We have $$ 1_\mathbb{Z}=ϕ(r)=ϕ(2(r/2))=2ϕ(r/2)$$  or equivalently that $ϕ(r/2)=\frac{1}{2}$. But this is not in $ \mathbb{Z}$, so there can be no such morphism.  Is my proof above correct? Thanks
111961
How do I show that $\text{Hom}(\mathbb{Z},G)$ and $G$ are isomorphic?  The method suggested by the book is to define a map $k$: $\text(\mathbb{Z},G) \to G$ by $f \to f(1)$.   I am stuck on how exactly to show that it is an isomorphism. Could anyone shed some light into this? Is $f(1)$ an element of G?  (For those who wanted to know, this is not homework)
115428
Let $R$ be a unital ring with cyclic additve group $(R, +,0)$. Is it the case that $1$ generates the additive group $(R,+,0)$?  Thoughts: Maybe classifying the unital rings with cyclic subgroups is possible.
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I am trying to construct a theoretical Diffie-Hellman key exchange protocol. However, i cannot understand the difference between choosing an additive group or a multiplicative group. I believe an additive would make it much more simple, but would a multiplicative group make it more difficult to break? What would be the difference between the use of those 2 kinds of number groups?
33550
Is there a better (or other) way(s) to prove the following statement? Also, the same argument works for multiplicative groups $\mathbb{R}-\{0\}$ and $\mathbb{Q}-\{0\}$, right?       Problem Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.   Solution By cantor's diagonal argument, there is no possible bijection between $\mathbb{Q}$ and $\mathbb{R}$. Since an isomorphism needs to be a bijection, there is no possible isomorphism between the additive groups $\mathbb{R}$ and $\mathbb{Q}$.    Thanks
7149
This is a very simple question but I believe it's nontrivial.  I would like to know if the following is true:      If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ are isomorphic.    Thanks!  If there isn't a proof (or disproof) of the general result, I would be interested to know if there are particular cases when this claim is true.
111625
While solving a problem on group theory, I encountered the quotient group Z x Z / &lt; (4, -6)>. Here Z is the integer.  At first I thought it is just Z/(4Z) x Z/(6Z). But I was wrong. the quotient group seems like something more trickier.  Could anyone show me to what other (more familiar group) is the one isomorphic to?
55550
My first instinct is to go to the 1st isomorphism theorem and say that the two rings are isomorphic if there is a field where $(x^2)=(x^2-1)$, which doesn't seem terribly promising.  But then it seems to me that $F[x]/(x^2)$ is the ring you get when you take $F$ and adjoin an element that satisfies the relation $x^2=0$, which would be $0$ and every field's got one.  Same with $F[x]/(x^2-1)$ which is $F$ adjoined by an element satisfying $x^2=1$.  $1$ does the trick and every field's got one of those too.  Any hints (or answers!) would be appreciated.  Thanks...