- #1
sairalouise
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If a is the only element of order 2 in a group, does it belong to the centre Z(G)?
Order of an element in relation to the centre.
In summary, the conversation discusses whether or not an element of order 2 in a group belongs to the centre Z(G). One person argues that there is always a certain special (and familiar) element of order 2 in every group, while another person disagrees and mentions the identity element with order 1. The conversation concludes with the acknowledgement that mistakes happen to everyone.
- #2
morphism
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Yes.
- #3
cup
- 27
- 0
Hint: In every group, there is always a certain special (and familiar) element, of order 2.
- #4
morphism
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Not true. If you're thinking of the identity element, then it has order 1.cup said:
I don't want to give the OP any hints because he/she hasn't posted any work.
- #5
cup
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morphism said:
You are right, of course. How embarrassing...
- #6
morphism
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It happens to all of us!cup said:
FAQ: Order of an element in relation to the centre.
What is the order of an element in relation to the centre?
The order of an element in relation to the centre is the number of times the element has to be multiplied by itself to get the identity element (usually denoted as e or 1). This is also known as the exponent of the element.
How is the order of an element related to the group it belongs to?
The order of an element is related to the group it belongs to in the sense that it must be a divisor of the order of the group. This means that the order of the element must be a factor of the total number of elements in the group.
What is the significance of the order of an element in relation to the centre?
The order of an element in relation to the centre is significant because it helps to determine the structure and properties of the group. It also helps in understanding the symmetry and patterns within the group.
Can an element have infinite order in relation to the centre?
No, an element cannot have infinite order in relation to the centre. This is because the order of an element must be finite and the group must have a finite number of elements.
How can the order of an element be calculated?
The order of an element can be calculated by repeatedly multiplying the element by itself until it reaches the identity element. The number of times the element is multiplied is its order. Alternatively, the order can also be calculated using mathematical formulas specific to the group.