• Jan 24th 2011, 10:56 PM
thanhuthe
Let G be an nonabelian has oder of 8. Let x be a character. Show that x(1)=1
• Jan 25th 2011, 12:48 AM
tonio
Quote:

Originally Posted by thanhuthe
Let G be an nonabelian has oder of 8. Let x be a character. Show that x(1)=1

A character is always a group homomorphism, so...

Tonio

Ps. By the way, the above is true for ANY group, not only non-abelian ones of order 8.
• Jan 25th 2011, 02:44 AM
TheArtofSymmetry
Do you mean a multiplicative character (linear character) or a character of a representation? If you are referring the latter, it is not always true that x(1)=1.

For example, $\displaystyle \phi:D_8 \rightarrow GL_2(\mathbb{Re})$ can be a matrix representation (verify this), which maps the identity of $\displaystyle D_8$ to $\displaystyle 2 \times 2$ identity matrix in $\displaystyle GL_2(\mathbb{Re})$, i.e., $\displaystyle \chi(1)=2$, where $\displaystyle \chi$ is the character of $\displaystyle \phi$.
• Feb 16th 2011, 05:18 AM
thanhuthe
Thanks so much!