Order of a Group

From (3): since H is not cyclic, it's order cannot be a prime number. Hence order of H cannlt be 7 and now we're left with one out of 1,2,4,8,56

How do I proceed further from here?

2 is a prime....

Find the order of a group H, given the following:

(1) H is a subgroup of some group G with order 168

(2) H is a subgroup of another group K with order 112

(3) H is not cyclic and dihedral

(4) H contains an element of order 7

(5) H has more than 2 left cosets in K

My work,

According to Lagranage theorem, we can conclude the following from (1) and (2)

From (1), since order of H divides the order of G(168), order of H can be one out of 1,2,3,4,7,8,12,14,21,24,32,56,64,168

From (2), since order of H divides the order of K(112), order of H can be one out of 1,2,4,7,8,16,28,56,112

From (1) & (2), order of should be one out of 1,2,4,7,8,56

You've got a few errors.

In your first list there are potentially 16 different orders from (1).

In your second list it should be 14, rather than 16 as one of the options.

Couple of things:

If a|b and a|c, then a | hcf(b,c). So use that with (1) and (2) to start off.

Then if you write your result as a product of primes, it should be easy to reel off all possibilities.

I'd look at (4) next, then (3) and (5).

Thanks for pointing that out.
But, what can we conclude from (4)? Does that implies order of H is a multiple of 7?

Yes, so that narrows it down to 4 possibilites. (Assuming you've only looked at (1) and (2)).

So, from 1 and 2 and since H is not cyclic, order of H can be one out of 1,4,8,14,28,56

From 4, order of H should be a multiple of 7. That is, order of H can be one out of 14, 28 and 56.

What can we conclude from 5 and H being dihedral?

Is H dihedral?

I'd interpret "(3) H is not cyclic and dihedral " to mean not cyclic and not dihedral. Can you upload an image of the original question?

I actually have the question written down.
Assuming it's not dihedral, what does it mean?

Google dihedral groups for info on that.

In this case consider a subgroup of order 14. What are the possible groups of order 14, in general.

For dihedral groups, I got something as below, which I don't understand.

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

To finish off and incorporate (5), you need to be familiar with some of the properties of cosets, which is something you should have covered since you mention Lagrange's theorem in your first post. Again, if you need more info there, youtube has several videos.

Apologies for the hijack - hopefully the OP is sorted now - but as someone who learned stuff without this fancy new-fangled internet thing, can I ask is there a "reliable" source of uni-level material on YouTube, i.e. is it just a case of searching for the appropriate topic and hoping for the best, or do you have a recommended "go to" list of good-quality lecture standard videos, or a specific channel you use?

I'm sure there are reliable sources, but I've not looked for them - I predate the internet. I just search on the appropriate topic, and if I come across a useful channel, I bookmark it for future reference, but then invariably never go back to it. Ho, hum!

Edit: Just checked some of the sites I'd bookmarked and the majority are now defunct. MIT have some online courses (which should be high quality), but for trying to find a particular topic I'd just do a search.