Show the dihedral group ##D_6## is isomorphic to ##S_3 x Z_2##

In summary, the conversation discusses various groups and their orders, including ##S_3##, ##Z_2##, ##S_3 x Z_2##, and ##D_6##. The group ##D_6## is defined as the symmetry group of a hexagon and can also be represented using generators and relations. The group operation chosen for all groups is multiplication. The group ##S_3 x Z_2## has 12 elements, the same number as the elements listed for ##D_6##, and a correspondence can be established between them while keeping the orders of the elements.
  • #1
Robb
225
8
Homework Statement
Show that the dihedral group ##D_6 is isomorphic to ##S_3 x Z_2## by constructing an explicit isomorphism. ( Hint: Color every other vertex of a hexagon red. Which elements of ##D_6## permute these 3 vertices. Also, which elements are order 2?)
Relevant Equations
##S_3 = { e, f, g, f^2, gf, gf^2 }##
##Z_2 = { 0,1 }##
##D_6 = { e, r, r^2, r^3, r^4, r^5, s, sr, sr^2, sr^3, sr^4, sr^5 }##
##f^3 = e##
##g^2 = e##
I'm not really sure where to begin with this problem. Any insight as to where to begin or what to look out for would be much appreciated!

Orders of ##S_3##
##|e|=1##
##|f|=3##
##|f^2|=2##
##|g|=2##
##|gf|=2##
##|gf^2|=3##

Orders of ##Z_2##
##|0|=1##
##|1|=2##

Orders of ##S_3 x Z_2##
##|e,0|=1##
##|e,1|=2##
##|f,0|=2##
##|f,1|=2##
##|f^2,0|=2##
##|F^2,1|=6##
##|g,0|=2##
##|g,1|=6##
##|gf,0|=3##
##|gf,1|=2##
##|gf^2,0|=2##
##|gf^2,1|=2##

Orders of ##D_6##
##|e|=1##
##|r|=6##
##|r^2|=3##
##|r^3|=2##
##|r^4|=3##
##|r^5|=6##
##|s|=2##
##|sr|=2##
##|sr^2|=2##
##|sr^3|=2##
##|sr^4|=2##
##|sr^5|=2##
 
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  • #2
What is ##D_6##? I'm asking since there are various definitions for it. The coloring hint indicates it is the symmetry group of a hexagon, the representation you gave under Relevant Equations indicates that you can use generators and relations, too.

Furthermore I would write ##S_3## explicitly with permutations rather than with ##e,f,g##, but that's only a side note. ##S_3=\langle e,f,g\,|\,e=f^3=g^2 \rangle## works as well. I also would choose only one group operation. As you have chosen the multiplication for ##D_6## and ##S_3##, let's do the same with ##\mathbb{Z}_2=\{\,-1,+1\,\}##.

Now ##S_3 \times \mathbb{Z}_2## as a set are all pairs ##(\sigma,\varepsilon)## with ##\sigma\in S_3## and ##\varepsilon \in \mathbb{Z}_2##. These are ##12## pairs total, same as ##12## elements you listed for ##D_6##. Have you tried to establish a correspondence between them? And if possible by keeping the orders of the elements!
 

FAQ: Show the dihedral group ##D_6## is isomorphic to ##S_3 x Z_2##

The 5 most frequently asked questions about "Show the dihedral group ##D_6## is isomorphic to ##S_3 x Z_2##" are:

What is the dihedral group ##D_6##?

The dihedral group ##D_6##, also known as the group of symmetries of a regular hexagon, is a mathematical group that consists of all the possible symmetries of a hexagon. It can be represented by a set of rotations and reflections.

What is the group ##S_3 x Z_2##?

The group ##S_3 x Z_2## is the direct product of the symmetric group ##S_3##, which is the group of permutations of 3 objects, and the cyclic group ##Z_2##, which is the group of integers modulo 2. This group is used to represent the symmetries of a hexagon in a different way.

What does it mean for two groups to be isomorphic?

Two groups are isomorphic if they have the same structure, meaning that they have the same number of elements and the same group operations. In other words, they are equivalent in terms of their mathematical properties.

How can you show that ##D_6## is isomorphic to ##S_3 x Z_2##?

To show that two groups are isomorphic, we need to find a bijective function, or a mapping, between the two groups that preserves their group structure. In this case, the mapping between the two groups can be defined by associating each rotation and reflection in ##D_6## with a permutation in ##S_3## and an element in ##Z_2##, respectively.

Why is it important to show that ##D_6## is isomorphic to ##S_3 x Z_2##?

Proving that these two groups are isomorphic allows us to understand the symmetries of a hexagon in a different way. It also helps us to understand the relationship between different mathematical structures and how they can be represented in different ways. This can have implications in various areas of mathematics and physics.

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