How many elements of order 50 are there in this group?

In summary, elements of order 50 in a group are those that, when raised to the 50th power, result in the identity element. The number of elements of order 50 in a group can be determined by using a formula or by finding the number of elements in each cyclic subgroup of the group. A group can have multiple elements of order 50, but there is a limit to the number which is determined by the total number of elements and the order of the elements. The structure of a group can greatly affect the number of elements of order 50 it has, with cyclic groups having a higher number compared to groups with more complex structures.
  • #1
feyomi
4
0
The group in question is U100, the group of units modulo 100, which, correct me if I'm wrong, is equal to {3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99}.

How many elements are there of order 50? How would I go about working this out, without having to explicitly calculate the order of each element?

Thanks.
 
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  • #2
You are missing one element, 1. The group has order 40, so there are no elements of order 50. Maybe you ask about elements a such that a^50 = 1, or equivalently a^10 = 1, which looks like being 20 elements, half the group.
 

1. What is the definition of "elements of order 50" in a group?

The order of an element in a group is the smallest positive integer n such that the element raised to the nth power is equal to the identity element. Therefore, elements of order 50 in a group are those that, when raised to the 50th power, result in the identity element.

2. How do you determine the number of elements of order 50 in a given group?

To determine the number of elements of order 50 in a group, you can use the formula:
Number of elements of order 50 = (number of elements in the group) / (number of elements in a cyclic subgroup of order 50). This can also be calculated by finding the number of elements of order 50 in each cyclic subgroup of the group and then adding them together.

3. Can a group have more than one element of order 50?

Yes, a group can have more than one element of order 50. In fact, a group can have any number of elements of order 50, as long as the total number of elements in the group is divisible by the order of the elements of order 50.

4. Is there a limit to the number of elements of order 50 in a group?

Yes, there is a limit to the number of elements of order 50 in a group. This limit is determined by the total number of elements in the group and the order of the elements of order 50. The maximum number of elements of order 50 that a group can have is equal to the total number of elements in the group divided by the order of the elements of order 50.

5. How does the structure of a group affect the number of elements of order 50?

The structure of a group can greatly affect the number of elements of order 50 it has. For example, if a group is cyclic, meaning all its elements can be generated by a single element, then the number of elements of order 50 will be equal to the number of elements in the group. On the other hand, if a group has a more complex structure, such as being a direct product of smaller groups, the number of elements of order 50 may be more limited.

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