Question

Prove: If a group has a pair of cyclic subgroups each of order 2, then it has a subgroup of order 4.

Answer 1

(c)
Find the order of each element of
G
. How are these orders arithmetically related
to the order of the group.
|
1
|
= 1
|
2
|
=
|
4
|
=
|
5
|
=
|
6
|
=
|
8
|
= 2
|
3
|
=
|
3
|
= 4
The orders of the elements, 2 and 4, are divisors of the order of the group, 8.
21.
Must the centralizer of an element of a group be Abelian?
No. For example, the centralizer of the identity is the entire group
G
, which is not
necessarily Abelian.
22.
Must the center of an group be Abelian?
Yes. Since an element of the center commutes with everything in the group, it must
commute with everything in the center, so the center is an Abelian subgroup.
Written Problems
10.
Prove that an Abelian group with two elements of order 2 must have a subgroup of
order 4.
Let
G
be an Abelian group and let
a, b
∈
G
with
a
6
=
b
and
|
a
|
=
|
b
|
= 2. We
want to show that
{
e, a, b, ab
}
is a subgroup of
G
of order 4. First we show it is closed
via a Cayley table:
e a b ab
e
e a b ab
a
a a
2
ab aab
b
b ba b
2
bab
ab
ab aba abb abab
e a b ab
e
e a b ab
a
a e ab b
b
b ab e a
ab
ab b a e
We simplify to the table on the right using
a
2
=
b
2
=
e
, since the order of both
a
and
b
are 2, and
ab
=
ba
, since
G
is Abelian.
By the Finite Subgroup Test, if a nonempty subset is closed, then it is a subgroup. So,
{
e, a, b, ab
}
6
G
.
It looks like our subgroup has order 4, but what if two elements of the set are ac-
tually the same? We know neither
a
nor
b
are the identity, since they both have order
2, and
e
always has order 1. If
a
=
ab
or
b
=
ab
, then
a
or
b
is the identity, which we
know if not true. So, could
ab
=
e
? If
ab
=
e
, then
a
−
1
=
b
, but
a
2
=
e
, so
a
−
1
also
equals
a
. Since inverses are unique, this means that
a
=
b
. Since we chose
a
and
b
to
be distinct, this is a contradiction. Thus, we have four distinct elements, so the order
of the subgroup is 4

Answer 2

hey what kind of math si this looks interesting... what are these cyclic groups meaning?

Answer 3

um its a set of number with an operation that can be generated(created) with only one element and its inverse
example:
integers under addition is cyclic
its called <1> so any integer can be found by repeated addition of 1 and its inverse -1 (additive inverse is not the same as multiplicative)

Answer 4

can you just paste the link to where ever you found that? @FutureMathProfessor

Answer 5

Dan, this stuff is definitely NOT interesting! lol

Answer 6

omg you just don't know yet.... its the most beautiful math.

Answer 7

hey tell me where i can start learning this stuff i wannntt too!!

Answer 8

u know any good places on line with this course lectures

Answer 9

many books, just pick one.
nope:( but im sure many.

Answer 10

@FutureMathProfessor give me link and ill give you medal....

Answer 11

@FutureMathProfessor you are doing group theory and you like it I assume, you just don't know you are doing group theory yet:)

Answer 12

hey also before i get start... expand on your statement why do u think this is the most beautiful mathematics

Answer 13

stuff like a^b*a^c = a^(b+c)

Answer 14

because there is no hand waving

Answer 15

Give me a sec

Answer 16

the algebra you do is operation on groups
or rings or fields

Answer 17

i watched like a video or 2 on sets before,, like silly stuff... proof for power sets and that stuff

Answer 18

i stopped watching.. but i can see how it can get interesting very soon

Answer 19

http://gyazo.com/b12ea84e00869a57832c0953eacbbe82.png

Answer 20

I have not done much with set theory

Answer 21

ty @FutureMathProfessor
after calculus its all fun, but I like abstract algebra.

Answer 22

NO. I actually liked Advanced Calculus better than Abstract Algebra, and that's an understatement.

Answer 23

you like chasing epsilons and deltas all day? yuk

Answer 24

jk

Answer 25

what sort of advanced calculus?

Answer 26

can you prove lim goes to infinity of 1/x @FutureMathProfessor ?

Answer 27

1/0 is infinity, so infinity is the answer

Answer 28

x goes to infinity

Answer 29

the limit is 0, but can you prove it?
if not I don't think we are talking about the same kind of advanced calculus, this is about as basic as it gets...

Answer 30

Yeah I meant 0. Sorry about that ;D

Answer 31

proof?

Answer 32

We must be.