Given a ser G and says is a group What elements have ord

Given a ser G = { } and says () is a group What elements have orders 2? Why? Toll me the elements in the cyclic subgtoup ? Why? Is a normal subgtoup? Why? Determine the subgtoup generated by {2, 3, 4, }? Why?

Solution

1) Order of element of a group : Order of element a is defined as n is any least positive integer such that a^n =e. (e=identity element),n is order of the element.

If no positive integer n such that a^n=e then we say that order of a =0 or infinite.

Here, 1 is the identity element.

a) No positive integer n such that 2^n=1

Therefore,

order of element 2 =0 or infinite.

b) No positive integer n such that 5^n=1

Therefore,

order of element 5 =0 or infinite.

2) If G is a cyclic group generated by a then it is denoted by G=<a>

4^1=4

4^2=16,which does not belongs to group G .

Therefore,

cyclic subgroup=<4>={4}.

4) Sub groups generated by {2,3,4} = {2,3} and {2,4}

2*3=6 which belongs to G

2*1=2

3*1=3

2*3=3*2=6.

Therefore {2,3} is a sub group of G

2*4=8 which belongs to G

2*1=2

4*1=4

2*4=4*2=8.

Therefore {2,4} is a sub group of G.

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