Question

Let G be a group of order p^2, where p is a prime.

Show that G must have a subgroup of order p.

please show with notation if possible

Answer #1

If G is a group of order (p^k)s where p is a prime number such
that (p,s)=1, then show that each subgroup of order p^i ; i=
1,2...(k-1) is a normal subgroup of atleast one subgroup of order
p^(i+1)

Let p,q be prime numbers, not necessarily distinct. If a group G
has order pq, prove that any proper subgroup (meaning a subgroup
not equal to G itself) must be cyclic. Hint: what are the possible
sizes of the subgroups?

: (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...

Let p be a prime. Show that a group of order
pa has a normal subgroup of order
pb for every nonnegative integer b ≤
a.

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)).

Supose p is an odd prime and G is a group and |G| = p 2 ^n ,
where n is a positive integer. Prove that G must have an element of
order 2

12.29 Let p be a prime. Show that a cyclic group of order p has
exactly p−1 automorphisms

Let G be a finite group and let H be a subgroup of order n.
Suppose that H is the only subgroup of order n. Show that H is
normal in G.
Hint: Consider the subgroup aHa-1 of G.
Please explain in detail!

Suppose N is a normal subgroup of G such that |G/N|= p is a
prime. Let K be any subgroup of G. Show that either (a) K is a
subgroup of N or (b) both G=KN and |K/(K intersect N)| = p.

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) =
xp
is a homomorphism
b. If G is
an Abelian group under addition, show that the mapping
f : G → G
f(x) = xpis a homomorphism.

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