Table of Contents
How do you know if UN is cyclic?
U(n) is cyclic if and only if n is 1, 2, 4,pk, or , 2pk, where p is an odd prime and k ≥ 1.
Is Z13 cyclic?
We have now counted all the elements of Z13* with order 6 or less, and there are exactly 8 of them. That leaves 4 elements unaccounted for, and these must therefore have order 12. Therefore Z13* has a generator (an element of order 12) and is cyclic.
Is Z24 cyclic?
Z24 is cyclic, there is exactly one subgroup for any divisor d of 24. The divisors are 1,2,3,4,6,8,12,24, so the answer is eight subgroups.
Is u30 cyclic?
<1>,<7>,<17>,<11>,<29>,<19>. Note that U(30) itself is not a cyclic group.
Is U 11 a cyclic group?
Note that the group U (11) is cyclic and that (2)=U(11). List the elements of U(11) and what operation makes U (11) a group.
Is 2 a generator of Z11?
We find the subgroups generated by group elements 2 and 5. We raise them to the powers 0,…,9. 2 is a generator and thus Z11 is cyclic.
What is the generator of z30?
The generator(s) of the group Z30 is/are. 3. 7.
Is U 40 a cyclic?
= 9 and U(40) is commutative, we see that {1,9,11,19} is closed under multiplication and thus a subgroup of U(40) of order 4. It is not cyclic since none of its elements has order 4.
Which of the following is not a cyclic group?
∴{1,3,5,7} under multiplication mod 8 is not a cyclic group.
How do you prove that a ring is cyclic?
U n is cyclic iff n is 2, 4, p k, or 2 p k, where p is an odd prime. The proof follows from the Chinese Remainder Theorem for rings and the fact that C m × C n is cyclic iff ( m, n) = 1 (here C n is the cyclic group of order n ).
Can the group U16 be cyclic?
Therefore, the group cannot be cyclic. Similarly, since 9 2 ≡ 1 ( mod 16) and 15 2 ≡ 1 ( mod 16), U 16 has at least two distinct elements of order 2, and therefore cannot be cyclic. In general, a cyclic group of order n has exactly ϕ ( d) elements of order d for each divisor d of n. Some shortcuts are available.
Is the group of order n cyclic?
A group of order n is cyclic if and only if it has an element of order n. Then by using the above theorem , this group is indeed not a cyclic group. Question : do I really have to check each element in the group for its order?
Do you have to check all elements in a cyclic group?
So yes, you must check ALL elements. As I mentioned yesterday in reply to a question of yours: a cyclic group has at most one element of order 2. You accepted that answer. Didn’t you read it before accepting it?