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Question: The symmetry group of a regular pentagon is a group of order 10. Show that it has subgroups of each of the orders allowed by Lagrange's theorem, and sketch the lattice of subgroups.

I got the subgroups:

Order 1: {identity}

Order 2: {identity and a reflection}

Order 5: {identity and 4 rotations}

Order 10: the whole group

How do I draw a lattice for these?

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How is the order of the lattice defined ? Is it by inclusion of groups? –  frabala Feb 4 at 14:32
Not quite sure what you are asking. Everything given to me is in the question. –  Bobby Lee Feb 4 at 14:36
The partial order by containment seems to be implied, and it's probably safe to assume. –  rschwieb Feb 4 at 14:56

3 Answers 3

up vote 3 down vote accepted

Normally Hasse diagrams of lattices are drawn so that "big things" are at the top, and a line between items indicates that there is no other node between those two items.

Here's a start (you'll have to complete it)

enter image description here

The dihedral group for the pentagon is not really very fun since there are so few divisors of $10$. You will get a much more interesting exercise if you try the dihedral group for the hexagon. I encourage you to try it out!

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By Lagrange, the subgroups of order $2$ and $5$ have only the identity element in common. I think this should allow you to draw the lattice.

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Thanks. So in order to draw this, I connect the subgroups that have common elements? –  Bobby Lee Feb 4 at 14:38
You connect the subgroups according to containment. Start with the whole group at the top. Draw lines to each of the subgroups of order $2$ and $5$. Moving down again, draw lines from each of these subgroups to the trivial subgroup. –  Peter Crooks Feb 4 at 14:43
@Lucas, I don't think so. The exercise is asking for the lattice of the subgroups. Two subgroups might have a common element (for example, all have common the identity element) but still not be subgroups with each other. I think you need to connect each groups to its subsgroups. –  frabala Feb 4 at 14:44
Yes, exactly. Do not connect subgroups unless one is contained in the other. –  Peter Crooks Feb 4 at 14:46

To sketch the lattice you will write down all the subgroups that you have found (including G and the empty group) and you'll draw a directed line from each subgroup to others that it can be included. G must turn out as the greates element of the lattice (so, all subgroups must have an edge towards G) and $\emptyset$ the least element (so, all the groups must have an edge from the empty set).

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There is no such thing as an "empty group." The axioms of a group include the line "there exists an element $e\in G$..." –  rschwieb Feb 4 at 19:11
Oups! That was a gaffe! –  frabala Feb 4 at 20:56

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