Question

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.

Answer #1

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.

1. Let a and b be elements of a group, G, whose identity element
is denoted by e. Assume that a has order 7 and that a^(3)*b =
b*a^(3). Prove that a*b = b*a. Show all steps of proof.

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

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)).

2. Let a and b be elements of a group, G, whose identity element
is denoted by e. Prove that ab and ba have the same order. Show all
steps of proof.

13.
Let a, b be elements of some group G with |a|=m and |b|=n.Show that
if gcd(m,n)=1 then <a> union <b>={e}.
18. Let G be a group that has at least two elements and has no
non-trivial subgroups. Show that G is cyclic of prime order.
20. Let A be some permutation in Sn. Show that A^2 is in
An.
Please give me steps in details, thanks a lot!

Suppose that a cyclic group G has exactly three subgroups: G
itself, e, and a subgroup of order p, where p is a prime greater
than 2. Determine |G|

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

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?

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)

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