Show <a in D8 : a^2=1> is not a group

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In summary, the conversation discusses the question of proving that <a in D8 : a2=1> is not a group, with different interpretations being considered. The proper interpretation is determined to be <a in D8 : a2=1> = <(a,b): a2=e, a4=e,b2=2, ab=a-1b>, making it equivalent to D4. The conversation also asks for help in interpreting the question by checking the order of elements in D8 and finding two elements of order 2 that do not produce another element of order 2 when multiplied together.
  • #1
Daniiel
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Hey,

I've been trying to solve this question,

Show that <a in D8 : a2=1> is a not a group.

I might not be processing it properly, but my interpretation of the question is that

<a in D8 : a2=1> = <(a,b): a2=e, b2=2, ab=a-1b>

Which is just D4, a group, the set of dihedral where 2n=4,

Is the proper interpretation supposed to be

<a in D8 : a2=1> = <(a,b): a2=e, a4=e,b2=2, ab=a-1b>

which is again D4,

Could anyone help me interoperate the question
 
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  • #2
Notice that D8 = {e,r,r2,r3,s,sr,sr2,sr3}. Now check which of these elements have order 2. Then find two elements of order 2 which when multiplied together do not produce an element of order 2.
 

FAQ: Show <a in D8 : a^2=1> is not a group

1. What does it mean for a set to be a group?

A group is a mathematical structure consisting of a set and an operation that satisfies four properties: closure, associativity, identity, and inverse. This means that for any elements in the set, the operation performed on them will result in another element within the set, the operation can be performed in any order, there exists an identity element that when operated on with any other element will result in that same element, and for every element there exists an inverse element that when operated on together will result in the identity element.

2. How is the set not a group?

The set does not satisfy the closure property. This means that when performing the operation of multiplication on two elements within the set, the result may not always be an element within the set. In this case, when multiplying any two elements

3. Can you provide an example to illustrate why is not a group?

Yes, for example, let's take the elements is not a group.

4. Are there any other properties that a set must satisfy to be considered a group?

Yes, in addition to the four properties mentioned before, a set must also satisfy the commutative property, meaning that the order in which the operation is performed does not change the result. It must also satisfy the cancellation property, meaning that if a * x = b * x, then a = b. These two properties are not satisfied by the set , further proving that it is not a group.

5. Can the set be made into a group by changing the operation?

No, the set cannot be made into a group by changing the operation. This is because the closure property will still not be satisfied with any other operation. In fact, any operation that is defined on this set will result in the same elements and therefore will not change the fact that the set does not satisfy the closure property.

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