# isomorphism

2. ### Group R^× isomorphic to the group R?

Question: Is the group R^× isomorphic to the group R? Why? R^× = {x ∈ R|x not equal to 0} is a group with usual multiplication as group composition. R is a group with addition as group composition.Is there any subgroup of R^× isomorphic to R? What I Know: Sorry, I would have liked to show some...
3. ### Show Isomorphism Between Set Real Numbers

Problem: Show that (R1, +1) is isomorphic to (Rc, +c) for any c > 0 by defining an explicit function ∅: R1 --> Rc and showing that it is an isomorphism. (Hints: 1 is not in R1, but if it were, what would ∅(1) be? What must ∅(0) be?) I know what are the requirements to prove the...
4. ### Prove Second Isomorphism Theorem for rings

Let $A$ be an ideal of a ring $R$ and let $S$ be a subring of $R$ . Show: (1) $S+A$ is a subring, (2) $A$ is an ideal of $S+A$ , (3) $S\cap A$ is an ideal of $S$ , (4) $(S+A)/A \cong S/(S\cap A)$. I can prove the first three. However, I'm not real confident of my construction of an...
5. ### Group Isomorphism Question

Let $G=\mathbb{R}-\{-1\}$ Define * on G by $a*b=a+b+ab$ Show that G is isomorphic to the multiplicitive group $\mathbb{R^x}$ OK, so I can show that G is an abelian group, and the text we're using says, without proof, that $\mathbb{R^x}$ is a group. So I want to use the following theorem...
6. ### Given phi is an isomorphism, prove that phi^-1 is an isomorphism as well

Prove that if $\phi :S \to S'$ is an isomorphism of $(S,*)$ with $(S',*')$ then $\phi^{-1}$ is an isomorphism of $(S',*')$ with $(S,*)$ aside: if $\phi$ is an isomorphism then we know it is one-to-one, onto and homomorphic. so now we know $\phi^{-1}$ exists because it is one-to-one...
7. ### First Isomorphism Theorem Applied to Dihedral Groups

Hello. I'm struggling a little bit with the first isomorphism theorem. I thought I was making progress, but then this question stumped me: "Let G be the dihedral group of order 10. If C denotes a cyclic group of order 2, use the first isomorphism theorem to find all homomorphisms from...
8. ### Groups and subgroups

So I have these two groups: G1 = ({1, 2, 4, 8, 16, 32, 43, 64}, ×85) G2 = ({0, 2, 4, 6, 8, 10, 12, 14}, +16) I've shown that they are both cyclicwith generators of 2,8,32, 43 for G1 and 2, 6, 10 for G2 but 1. I need determine an isomorphism between them 2. I need to determine all of the...
9. ### How to prove that T−2I is an isomorphism?

I'm having a difficulty with the following assignment, and I would be grateful if some one could help me. Given: T:R3→R3,T(1,2,3)=(−2,−4,−6),dimImT<dimkerT I need to prove: T−2I is an isomorphism. I know that -2 is an eigenvalue and that it is the only nonzero eigenvalue because dimImT=1, but...
10. ### Proving isomorphism in lattices

The question is as follows: Let $f$ be a monomorphism from a lattice $L$ to a lattice $M$.Show that $L$ is isomorphic to a sublattice of $M$. My attempt: Since $f$ is a monomorphism from a lattice $L$ to a lattice $M$ therefore it will be a lattice homomorphism which is injective. Since, $f$...
11. ### Not an isomorphism

I need to show that \mathbb{Q} \times Z_2 is not isomorphic to \mathbb{Q}. Mind you, I'm in the section of my book dealing with cyclic groups. I was able to handle the other problems in this set by showing that one group is cyclic and the and the other isn't, etc. But \mathbb{Q} isn't cyclic...
12. ### Generalized proof of isomorphism: Once more unto the breach!

I'm going to have to post the full form of this question as I doubt the terminology is universal. So here it goes! Okay, after that mouthful I am going to sketch the method I used to answer the question. I found the stabilizer, then calculated the order of each element. I did this also for...
13. ### Isomorphism

I have five problems from my text that I did exactly the same way. Thus I have reason to believe that I am doing something wrong. (Doh) The problem is showing that two groups are not isomorphic. To give you the flavor of what I am doing consider the following: It is clear to me that we can...
14. ### Direct Sum and Isomorphism

If, A,B, C are finite abelian groups then prove (i) A direct sum B isomorphic to A direct sum C implies that B is isomorphic to C (ii) A direct sum A isomorphic to B direct sum B implies A isomorphic to B.
15. ### Isomorphism

If A' is isomorphic to A'' and B' is isomorphic to B'' then A'UB' is isomorphic to A''UB''. (their intersections are empty). Moreover conclude that the disjoint union is well-defined up to isomorphism.
16. ### Proving Isomorphism

THe problem is attached. Anyone have any ideas?
17. ### Let G and H be the following subgroups of GL(2; R):

Let G and H be the following subgroups of GL(2, R): G ={ [a b,0 1 ] | a and b are in R and a is not equal to 0} and H={ [ 1 b ,0 1 ] | b is in R} : (a) Prove that H is normal in G. (b) Use the First Isomorphism Theorem to prove that G=H is isomorphic to R the group of nonzero real...
18. ### Cube. Coloring. Isomorphism.

Let G be a group of rigid motions of cube. a) Show that G = S_4 b) Show that the alternating subgroup A_4 \le S_4 is isomorphic to the group of rigid motions of regular tetrahedron. c)Find cycle index for both S_4 and A_4. d) Determine pattern inventory of m coloring of a set X =...
19. ### Isomorphism of Direct product of groups

find which of the following groups is isomorphic to S3 \bigoplus Z2. a) Z12 b) A4 c) D6 d) Z6 \bigoplus Z2 I eliminate option a because Z12 is cyclic whereas S3 \bigoplus Z2 is not because we know that the External direct product of G and H is cyclic if and only if the orders of G and H...
20. ### Problem on Quotient Groups and First Isomorphism Theorem

Q: Let P be a partition of a group G with the property that for any pair of elements A, B of the partition, the product set AB is contained entirely within another element C of the partition. Let N be the element of P which contains 1. Prove that N is a normal subgroup of G and that P is the set...