Question

Let G be a group of order p^3. Prove that either G is abelian or its center has exactly p elements.

Answer #1

Let G be a non-abelian group of order p^3 with p prime.
(a) Show that |Z(G)| = p. (b) Suppose a /∈ Z(G). Show that
|NG(a)| = p^2 .
(c) Show that G has exactly p 2 +p−1 conjugacy classes (don’t
forget to count the classes of the elements of Z(G)).

Let G be a group (not necessarily an Abelian group) of order
425. Prove that G must have an element of order 5. Note, Sylow
Theorem is above us so we can't use it. We're up to Finite Orders.
Thank you.

Let G be an Abelian group and H a subgroup of G. Prove that G/H
is Abelian.

Let H be a normal subgroup of G. Assume the quotient group G/H
is abelian. Prove that, for any two elements x, y ∈ G, we have x^
(-1) y ^(-1)xy ∈ H

Let G be an abelian group, let H = {x in G | (x^3) = eg}, where
eg is the identity of G. Prove that H is a subgroup of G.

Let G be a group of order 4. Prove that either G is generated by
a single element or g^2 =1 for all g∈G.

: (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.
(For the identity, remember that 1 = p 0 is a power of p.) (b) Let
p1, . . . , pn be pair-wise distinct primes, and let G be an
Abelian group. Show that Gp1 , . . . , Gpn form direct sum in...

LetG be a group (not necessarily an Abelian group) of order 425.
Prove that G must have an element of order 5.

Let G be a group of order 4. Prove that either G is cyclic or it
is isomorphic to the Klein 4-group V4 =
{1,(12)(34),(13)(24),(14)(23)}.

Let G be an Abelian group and let H be a subgroup of G Define K
= { g∈ G | g3 ∈ H }. Prove that K is a subgroup of G
.

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