Explanation Consider the statement, “The kernel of any homomorphism from group G to group G ' is a normal subgroup of G ' ”. Counter example: Let ϕ be the mapping from S n to the additive group ℤ 2 defined by ϕ ( f ) = { [ 0 ] i f f i s a n e v e n p e r m u t a t i o n [ 1 ] i f f i s a n o d d p e r m u t a t i o n To prove ϕ is a homomorphism, show that ϕ ( f ⋅ g ) = ϕ ( f ) + ϕ ( g ) for all f , g ∈ S n . Let f , g ∈ S n . f ⋅ g ϕ ( f ) + ϕ ( g ) ϕ ( f ⋅ g ) f , g both even even [ 0 ] + [ 0 ] = [ 0 ] [ 0 ] One even, one odd odd [ 0 ] + [ 1 ] = [ 1 ] or [ 1 ] + [ 0 ] = [ 1<

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Chapter 4.6, Problem 2TFE

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Whether the statement, “The kernel of any homomorphism from group G to group G' is a normal subgroup of G'” is true or false.

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Whether the statement, “The kernel of any homomorphism from group G to group G ' is a normal subgroup of G ' ” is true or false.

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Whether the statement, “The kernel of any homomorphism from group G to group G' is a normal subgroup of G'” is true or false.

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