A group G has exactly 8 elements or order 3

In summary, in a group G with exactly 8 elements of order 3, there are 3 different ways to get factors, resulting in 3 subgroups of order 3. Each subgroup has 3 elements of order 3, and two distinct subgroups of order 3 have no common elements of order 3.
  • #1
nowimpsbball
15
0
A group G has exactly 8 elements of order 3 (Unanswered as of 1/31)

How many subgroups of order 3 does G have?

So we have 8 elements, its prime decomposition is 8=2^3. The number of different ways to get factors is how many subgroups, at least that is what I interpret from my notes...so there are 2^3, 2*2^2, and 2*2*2, so three different ways to get factors, am I doing this right?

Thanks
 
Last edited:
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  • #2
Do you mean G has 8 elements OR order 3, or do you mean G has 8 elements OF order 3.
 
  • #3
d_leet said:
Do you mean G has 8 elements OR order 3, or do you mean G has 8 elements OF order 3.

g has 8 elements OF order 3, my bad
 
  • #4
every subgroup of order three has how many elements of order three?

and how many common elements of order three do two distinct subgroups of order three have?
 

1. What does it mean for a group to have exactly 8 elements?

It means that the group contains exactly 8 distinct elements, and no more or less. Each element in the group is unique and cannot be duplicated.

2. What is the order of a group?

The order of a group refers to the number of elements in the group. In this case, the group has an order of 8, meaning it contains 8 distinct elements.

3. How can a group have an order of 3 if it has 8 elements?

The order of a group does not refer to the number of elements in the group, but rather the number of elements that are required to generate the entire group through multiplication. In this case, only 3 of the 8 elements are needed to generate the rest of the group.

4. Is it possible for a group to have an order of 3 and more than 8 elements?

No, if a group has an order of 3, it can only have 3 distinct elements. However, the elements in the group may have different combinations and arrangements, but the group will still only have 3 elements overall.

5. What are some examples of groups with exactly 8 elements and order 3?

Some examples include the group of symmetries of an equilateral triangle, the group of symmetries of a square, and the group of rotations and reflections of a regular octagon.

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