Timeline for Zero divisors with support of size 3 in group algebras of finite groups

6 events

when toggle format	what		by	license	comment
Sep 5, 2015 at 13:13	comment	added	Noam D. Elkies		One way to explain this example is starting from an identity $g+g^2+g^4=0$ in the field of $2^3$ elements, which contains an element $g$ of multiplicative order $7$. There are similar identities from other finite fields or characteristic $2$; for example, $x^5 + x^2 + 1$ is irreducible mod $2$ so $g^5 + g^2 + 1$ is a zero-divisor when $g$ has exponent $31$.
Sep 5, 2015 at 9:47	comment	added	Alireza Abdollahi		@NoamD.Elkies: Have you any idea to generalize this example?
Sep 1, 2015 at 18:43	comment	added	Noam D. Elkies		Yes: let $G = \langle g \mid g^7 = 1 \rangle$ and $\alpha = g + g^2 + g^4$; then $\alpha (\alpha+1) = 0$.
Aug 31, 2015 at 5:49	comment	added	Alireza Abdollahi		Many thanks. This perfectly and very simply answer my question. Actually the answer shows all elements in the augmentation ideal of the group algebra whose support is of size 3 can be chosen as a candidate for the answer over any field with more than 2 elements. The construction may interpret as canonical as it is valid over any finite group and any field with more than 2 elements. Can one find a zero divisor with support of size $3$ over $GF(2)$? anyway again many thanks for your answer.
Aug 31, 2015 at 5:42	vote	accept	Alireza Abdollahi
Aug 30, 2015 at 10:28	history	answered	Ilya Bogdanov	CC BY-SA 3.0