# quick group theory Q

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#1
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15 years ago
#2
All subgroups of a cyclic group are cyclic.

A generator for such a subroup must have an order dividing 14.

The orders of the elements 0,...13 are

1,14,7,14,7,14,7,2,7,14,7,14,7,1 4,

So the subgroups are

{0}
{0,7}
{0,2,4,6,8,10,12}
C_14
0
#3
stil dont quite understand..
0
15 years ago
#4
stil dont quite understand..
Can you be more specific?
0
#5
what do u mean by generator..and i thort c14 was a group {e,x_1,x_2.....x_13} where e is the identity
0
15 years ago
#6
what do u mean by generator..and i thort c14 was a group {e,x_1,x_2.....x_13} where e is the identity
A generator is an element which, when you take all its powers, gives you the whole group. So a group is cyclic if and only if it has a single generator.

C_14 can be thought of a {e, g, g^2, ... , g^13} where g is a generator and g^14 = e. Another generator for the group is g^3 because its powers are

e, g^3, g^6, g^9, g^12, g, g^4, g^7, g^10, g^13, g^2, g^5, g^8, g^11,

But C_14 can also be thought of as the group of integers mod 14 - which is how my original was originally phrased - because it's easier to write down. So instead of 7 think g^7 etc.
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