How Do Generators {2, 3} Form Z6?

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In summary, <{1}> = <{5}> = <{2,3}> = Z6. This means that any subgroup containing 2 and 3 must also contain 5, except for {2,4} which does not generate Z6.
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ma3088
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I understand how {1} and {5} are generators of Z6.

{1} = {1, 2, 3, 4, 5, 0} = {0, 1, 2, 3, 4, 5}
{5} = {5, 4, 3, 2, 1, 0} = {0, 1, 2, 3, 4, 5}

But my book also says that {2, 3} also generates Z6 since 2 + 3 = 5 such as {2,3,4} and {3,4} I believe. Thus every subgroup containing 2 and 3 must also be 5 except for {2,4}.

Can someone explain this to me? Ty in advance.
 
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Your notation is incorrect. {1} is a set containing one element, 1. {1, 2, 3, 4, 5, 0} is a set containing six elements. Therefore {1} does not equal {1, 2, 3, 4, 5, 0}.

The usual notation for the group generated by a set is a pair of angle brackets: <{1}> denotes the group generated by the set {1}. It is true that <{1}> = <{5}> = Z6. It is also (trivially) true that <{1, 2, 3, 4, 5, 0}> = Z6.

Note that in general, if S is a subset of Z6, <S> is the smallest subgroup of Z6 which contains all of the elements of S. If S is a subgroup, then S = <S>. Also, it's easy to verify that if S [itex]\subseteq[/itex] T, then <S> [itex]\subseteq[/itex] <T>.

Now what about <{2,3}>? This is a group, by definition, so it must be closed under addition. Thus <{2,3}> must contain 5 because 2+3=5. In other words, {5} [itex]\subseteq[/itex] <{2,3}>. Therefore Z6 = <{5}> [itex]\subseteq[/itex] <{2,3}>. For the reverse containment, we have {2,3} [itex]\subseteq[/itex] {1,2,3,4,5,0}, so <{2,3}> [itex]\subseteq[/itex] <{1,2,3,4,5,0}> = Z6. We conclude that <{2,3}> = Z6.
 

FAQ: How Do Generators {2, 3} Form Z6?

1. What is a generator of Z6?

A generator of Z6 is an element that, when combined with itself a certain number of times, can produce all the other elements in the group. In Z6, the generators are 1 and 5.

2. How do you find generators of Z6?

To find generators of Z6, you can use the formula a^k mod n, where a is an integer and n is the size of the group (in this case, 6). If the resulting values are all unique, then a is a generator of Z6. For example, when a=1, we get the sequence 1, 1, 1, 1, 1, 1. When a=5, we get the sequence 5, 1, 5, 1, 5, 1, which are all unique values, making 5 a generator of Z6.

3. How many generators does Z6 have?

Z6 has two generators - 1 and 5. This is because Z6 is a cyclic group, meaning it can be generated by a single element.

4. What is the order of a generator in Z6?

The order of a generator in Z6 is the number of times the generator must be combined with itself to produce all the other elements in the group. In this case, since the generators are 1 and 5, their order is 6, as they must be combined with themselves 6 times to produce all the elements in Z6.

5. Can you have a generator of Z6 that is not 1 or 5?

No, in Z6, only 1 and 5 can be generators. This is because the order of any element in a cyclic group must be equal to the size of the group. Since Z6 has a size of 6, any other element in Z6 will have an order less than 6 and therefore cannot be a generator.

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