Group Theory

Question

Assume g is an element of the group.consider the following conditions of g with e as identity element.

1.g⁸=e

2.g²$\neq$e

3.Order of g is not 8.

Find the order of g.

Answer 1

Order of an element is smallest positive integer x,such that ${a^x}=e$

As given  ${a^8}=e$ ,it means order of an element cannot be more than 8.

8 is not the order and it is not identity element so order 1 and 8 order are not possible.Also,2 is not the order.

We are left with 3,4,5,6,7

As ${a^8}=e$ ,so order must be some multiple of 8,otherwise we will not get ${a^8}=e$

so only option which satisfy is 4

hence,4 is the order of g

Comments

g^8 may be any number of identity elements with operation of the group e^k=g^8
This k may be 1 which is given its not

 or 3 but can't be since we can't factor 8 into 3 equal units

neither 4 since g^2 isn't e
Similarly 5 6 and 7 equal units aren't possible hence 2
Which tells us the minimum number(4) which makes g^4=e

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Why 1 can't be the order

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if 1 would have been the order then $g^{2}≠e$ will not be satisfied.