What is Z8 isomorphic to?

The element [1]8 of Z8 has order 8. (b) The prime factorisation of 8 is 8 = 23, so by the FTAG, every abelian group of order 8 is isomorphic to Z23 or Z2 × Z22 or Z2 × Z2 × Z2, and these groups aren’t isomorphic.

Is q8 isomorphic to Z8?

Absolutely none. Z15 elements all have finite order. The only element of Z that has finite order is 0. Homomorphisms map elements of finite order to elements of finite order.

Can there be a Homomorphism from Z4 Z4 to Z8?

– Can there be a homomorphism from Z4 ⊕ Z4 onto Z8? No. If f : Z4 ⊕ Z4 −→ Z8 is an onto homomorphism, then there must be an element (a, b) ∈ Z4 ⊕ Z4 such that |f(a, b)| = 8. This is impossible since |(a, b)| is at most 4, and |f(a, b)| must divide |(a, b)|.

Is G H isomorphic to Z4 or Z2 Z2?

Is G/H isomorphic to Z4 or Z2 × Z2? How about G/K? Solution: G/H has 4 elements consisting of H, (1, 0) + H, (0, 1) + H and (1, 1) + H. The last three cosets have order 2, and hence G/H is isomorphic to the Klein group Z2 × Z2.

Is Z4 subgroup of Z8?

The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z4. is the group direct product of Z8 and Z2, written for convenience using ordered pairs with the first element an integer mod 8 (coming from cyclic group:Z8) and the second element an integer mod 2. The addition is coordinate-wise.

Is u16 isomorphic to Z8?

To be isomorphic to Z8, U(16) must have an element of order 8. To be isomorphic to Z4 ⊕ Z2, it must have only elts of order 1,2, or 4. To be isomorphic to Z2 ⊕ Z2 ⊕ Z2, it must have elts only of order 1 or 2. Thus U(16) ≈ Z4 ⊕ Z2.

Is U 10 and Z4 isomorphic?

(a) The mapping φ : Z4 → U(10) given by φ(0) = 1, φ(1) = 3, φ(2) = 9 and φ(3) = 7 is an isomorphism as the table suggests. Thus Z4 ≈ U(10).

Is U 16 isomorphic to Z8?

Can there be a homomorphism from Z8 ⊕ Z2 onto Z4 ⊕ Z4 give reasons for your answer?

Solution: These groups have the same order (16), so an onto homomor- phism would be a one-to-one homomorphism, and would have to be an isomorphism. However, Z8 ⊕ Z2 has an element of order 8, and Z4 ⊕ Z4 does not have any element of order 8, so the two groups are not isomorphic.

What are the subgroups of Z8?

(Subgroups of a finite cyclic group) List the elements of the subgroups generated by elements of Z8. 〈0〉 = {0}, 〈2〉 = 〈6〉 = {0, 2, 4, 6}, 〈4〉 = {0, 4}, 〈1〉 = 〈3〉 = 〈5〉 = 〈7〉 = {0, 1, 2, 3, 4, 5, 6, 7}.

What are the generators of Z8?

For k ∈ Z8, gcd(8,k)=1 if and only if k = 1,3,5,7. So there are four generators. Finally, for k ∈ Z20, gcd(20,k)=1 if and only if k = 1,3,7,9,11,13,17,19.