Question

Prove that if p is prime then Z_{p, where it is nonzero, is a
group under multiplication.}

Answer #1

Prove that if p is prime then Zp, where it is nonzero, is a
group under multiplication.

Please solve in full detail! Use the fact that
Zp*, the nonzero residue classes modulo a
prime p, is a group under multiplication to establish Wilson’s
Theorem.

Prove that the nonzero elements of a field form an abelian group
under multiplication

Suppose S is a ring with p elements, where p is prime.
a)Show that as an additive group (ignoring multiplication), S is
cyclic.
b)Show that S is a commutative group.

For any prime number p use Lagrange's theorem to show that every
group of order p is cyclic (so it is isomorphic to Zp

1(a) Suppose G is a group with p + 1 elements of order p , where
p is prime. Prove that G is not cyclic.
(b) Suppose G is a group with order p, where p is prime. Prove
that the order of every non-identity element in G is 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.

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

Show that if a square matrix K over Zp ( p prime) is
involutory ( or self-inverse), then det K=+-1
(An nxn matrix K is called involutory if K is invertible and
K-1 = K)
from Applied algebra
show details

Prove that the quotient group GLn(R)/SLn(R) is isomorphic to the
group R∗ (under multiplication).

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