Question

Let G be a group and let *p* be a prime number such that
*pg* = 0 for every element *g* ∈ G.

a. If G is commutative under multiplication, show that the mapping

*f* : G → G

*f*(x) =
*x ^{p}*

is a homomorphism

b. If G is an Abelian group under addition, show that the mapping

*f* : G → G

*f*(x) = *x ^{p}*is a homomorphism.

Answer #1

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: (a) Let p be a prime, and let G be a finite Abelian group.
Show that Gp = {x ∈ G | |x| is a power of p} is a subgroup of G.
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Abelian group. Show that Gp1 , . . . , Gpn form direct sum in...

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Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G →
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please show with notation if possible

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Let p be an prime. Use the action of G =
GL2(Z/pZ) on (Z/pZ)2 and the orbit-stabilizer
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(Given an element x ∈ X, the orbit O(x) of x is the
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This is a subgroup of G, and the orbit-stabilizer
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