* Click the radio button in front of "Cyclic group"
* Enter your "n" value for Zn.
* Press "Display Cayley Table" button to display the Cayley Table corresponding to your Zn group.
* Press "Display Subgroups" to display all the distinct subgroups in Zn
* Enter an element of your Zn group to determine whether or not it acts as a generator for your Zn group.
* Enter an element of your Zn group to find it's inverse.
* Click the radio button in front of "Dihedral group"
* Enter your "n" value for Dn. Remember, n must be 6, 8, or 12.
* Press "Display Cayley Table" button to display the Cayley Table corresponding to your Dn group.
* Press "Display Subgroups" to display all the distinct subgroups of Dn.
* Enter an element of your Dn group to determine whether or not it acts as a generator for your Dn group.
* Enter an element of your Dn group to find it's inverse..
NOTE: Currently, the tests for subgroups,
generators, and inverses
are unavailable. See Prof. Dietz for possible enhancements to
this project (if you are interested!)
* Click the radio button in front of "U-group"
* Enter your "n" value for Un.
* Press "Display Cayley Table" button to display the Cayley Table corresponding to your Un group.
* Press "Display Subgroups" to display all the distinct subgroups of Un.
* Enter an element of your Un group to determine whether or not it acts as a generator for your Un group.
* Enter an element of your Un group to find it's inverse..
NOTE: If you enter a non element of your
group, the
system won't catch the error. Instead, a "non-inverse" will be
displayed as well as incorrect generated elements. Please
only enter elements of your group to ensure proper results!
* Click the radio button in front of "Zn x Zm"
* Enter your "n" value for Zn and "m" value for Zm.
* Press "Display Cayley Table" button to display the
Cayley Table
corresponding to your Zn x Zm group.
* Not entering an element of your group. Check your cayley table for possible group elements.
* Forgetting to enter all group information.
1. Cayley button requires the group radio button to be selected before the cayley table button is pressed.
2. Cayley button requres the "n" value for the group to be entered before the cayley table button is pressed.
3. Generator button requires the group to be selected (the radio button is filled), the "n" value for the group to be entered, and an element of the group to be entered.
4. Inverse button requires the group to be selected, the "n" value for the group to be entered, and an element of the group to be entered.
5. Subgroup button requires the group to be selected, and the "n" value for the group to be entered.
6. The Zn x Zm Cayley button requires that
group to be selected
and the values for both "n" and "m" to
be entered.
Zn:
Zn = {0, 1, ... , n-1} under addition mod n.
Un:
U(n) is the set of positive integers less than and relatively
prime to n, under multiplication mod n.
Dn:
Dn is the dihedral group of order n. Group of symmetries
of a regular polygon with n/2 sides.
Zn x Zm:
Zn x Zm is the external direct product of two cyclic groups.
Subgroup
A nonempty subset, H, of a group G is called a subgroup of G
if, relative to the product in G, H itself forms a group.
Generators
We call an element a generator if it can "generate" all other
elements in the group.
Inverses
For every a in G there exists an element b in G such that a*b = b*a = e.
We write this element b as a^-1 and call it the inverse.