Question

a) Give an example of a group of order 360 that contains no subgroups of order 180, and explain why

b) Let G be a group of order 360, Does G have an element of order 5? please explain

Answer #1

That's easy.

You always need to think of group of permutations for such problems.

Have a close look and thumb up.

Have a great day!!!

Let G be a group with subgroups H and K.
(a) Prove that H ∩ K must be a subgroup of G.
(b) Give an example to show that H ∪ K is not necessarily a
subgroup of G.
Note: Your answer to part (a) should be a general proof that the
set H ∩ K is closed under the operation of G, includes the identity
element of G, and contains the inverse in G of each of its
elements,...

2.6.22. Let G be a cyclic group of order n. Let m ≤ n be a
positive integer. How many subgroups of order m does G have? Prove
your assertion.

A group G is a simple group if the only normal subgroups of G
are G itself and {e}. In other words, G is simple if G has no
non-trivial proper normal subgroups.
Algebraists have proven (using more advanced techniques than
ones we’ve discussed) that An is a simple group for n ≥ 5.
Using this fact, prove that for n ≥ 5, An has no subgroup of
order n!/4 .
(This generalizes HW5,#3 as well as our counterexample from...

Is it possible for a group G to contain a non-identity element
of finite order and also an element of infinite order? If yes,
illustrate with an example. If no, give a convincing explanation
for why it is not possible.

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.

Can you show an example of subgroups? and how to solve
them
I don't understand closure
If a,b, is an element in H then ab element in H (does ab mean
multiple)
I don't understand inverse
ab^-1 element in H. (if we have ab would the inverse be
a^-1b^-1)
Some example are much need something visual

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!

(a) Prove or disprove: Let H and K be two normal subgroups of a
group G. Then the subgroup H ∩ K is normal in G. (b) Prove or
disprove: D4 is normal in S4.

Give an example of a nontrivial subgroup of a multiplicative
group R× = {x ∈ R|x ̸= 0}
(1) of finite order
(2) of infinite order
Can R× contain an element of order 7?

Suppose that G is abelian group of order 16, and in computing the
orders of its elements, you come across an element of order 8 and 2
elements of order 2. Explain why no further computations are needed
to determine the isomorphism class of G. provide explanation
please.

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