1. E

    Proof for isomorphism

    I want to solve this. Please help. Show that (S,+) and (P, *) are isomorphic.
  2. M

    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. M

    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. O

    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. B

    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. J

    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. Wander

    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. D

    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. D

    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. M

    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. topsquark

    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. topsquark

    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. topsquark


    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. J

    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. J


    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. N

    Proving Isomorphism

    THe problem is attached. Anyone have any ideas?
  17. L

    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. V

    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. M

    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. A

    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...