What is a non-trivial unit in the integral group ring Z[S_3]?

  • MHB
  • Thread starter Euge
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    2016
In summary, a non-trivial unit in the integral group ring Z[S_3] is an element that is both invertible and not equal to 1. To determine if an element in Z[S_3] is a non-trivial unit, you can check if it has a multiplicative inverse by finding the greatest common divisor of the element and the order of the group. Z[S_3] can have multiple non-trivial units due to the fact that the group S_3 has more than one element of order 2. These units have significance in representing symmetries and in mathematical calculations and proofs. Non-trivial units in Z[S_3] are not unique as there can be multiple elements with a multiplicative inverse
  • #1
Euge
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Here is this week's POTW:

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Give an example of a unit of the integral group ring $\Bbb Z[S_3]$ that is not of the form $1x$ for some $x\in S_3$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
No one answered this week's problem. You can read my solution below.
Let $a = (12)$ and $b = (123)$. Then $a^2 = b^3 = 1$ and $ab = b^2 a$. Now define

$$g = 1 + a - b^2 + ab - ab^2,$$

$$h = -1 + a + b + ab - ab^2.$$

Then

$$gh$$

$$= -1 + a + b + ab - ab^2 - a + b^2 + ab + b^2 a - a^2 b^2 + b^2 - b^2 a - b^3
- b^2 ab + b^2 ab^2 - ab + aba + ab^2 + ab ab - ab ab^2 + ab^2 - ab^2 a - ab^3 - ab^2 ab + ab^2 ab^2$$

$$= -1 + a + b + ab - ab^2 - a + 1 + ab + b - b^2 + b^2 - ab - 1 - ab^2 + a - ab + b^2 + ab^2 + 1 - b + ab^2 - b - a - b^2 + 1$$

$$= 1$$

and

$$hg$$

$$= -1 - a + b^2 - ab + ab^2 + a + a^2 - ab^2 + a^2 b - a^2 b^2 + b + ba - b^3 + bab - bab^2 + ab + aba - ab^3 + abab - abab^2 - ab^2 - ab^2 a + ab^4 - ab^2 ab + ab^2 ab^2$$

$$= -1 - a + b^2 - ab + ab^2 + a + 1 - ab^2 + b - b^2 + b + ab^2 - 1 + a - ab + ab + b^2 - a + 1 - b - ab^2 - b + ab - b^2 + 1$$

$$= 1.$$

This shows that $g$ and $h$ are multiplicative inverses of each other, neither of which takes the form $1x$ for some $x\in S_3$.
 

Related to What is a non-trivial unit in the integral group ring Z[S_3]?

What is a non-trivial unit in the integral group ring Z[S_3]?

A non-trivial unit in the integral group ring Z[S_3] is an element that is both invertible and not equal to 1. In other words, it is an element that has a multiplicative inverse and is not the identity element.

How do you determine if an element in Z[S_3] is a non-trivial unit?

To determine if an element in Z[S_3] is a non-trivial unit, you can check if it has a multiplicative inverse. This can be done by finding the greatest common divisor of the element and the order of the group, which in this case is 6. If the greatest common divisor is 1, then the element is a non-trivial unit.

Can Z[S_3] have multiple non-trivial units?

Yes, Z[S_3] can have multiple non-trivial units. This is because the group S_3 has more than one element of order 2, which means there are more than one element with a multiplicative inverse in Z[S_3].

What is the significance of non-trivial units in Z[S_3]?

Non-trivial units in Z[S_3] can be used to represent symmetries in group theory. They can also be used in various mathematical calculations and proofs involving group rings.

Are non-trivial units in Z[S_3] unique?

No, non-trivial units in Z[S_3] are not unique. This is because there can be multiple elements with a multiplicative inverse in Z[S_3]. However, each non-trivial unit is unique within its own group, meaning that no two non-trivial units will have the same inverse within the group S_3.

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