Is {π ∈ S_n | sgn π = 1} a Subgroup of S_n?

Signum function sgn(x)=-1 if x<0

=0 if x=0

=+1 if x>0

Other defintion is sgn(x)=|x|/x for any real number x.

Given function is sgn pi

S_n=sgn pi

=+1 (since pi>0)

S_n={-1,0,+1} is a group since it satisfies the properties of closure,associativity,identity and invertibility.

Subgroups are subsets which should form a group under the same operation.i.e 3 conditions should be satisfied.

sgn pi is a subset of S_n since its value 1 is an element of S_n

sgn pi is a group since it involves the set <R,+> and S_n also includes the set <R,+>.

Hence the answer is yes its a subgroup of S_n.

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