Elementary Proofs thatZp2andZp3are CI-groups

Quotients of CI-Groups are CI-Groups

We show that a quotient group of a CI-group with respect to (di)graphs is a CI-group with respect to (di)graphs. In [1,2], Babai and Frankl provided strong constraints on which finite groups could be CI-groups with respect to graphs. As a tool in this program, they proved [1, Lemma 3.5] that a quotient group G/N of a CI-group G with respect to graphs is a CI-group with respect to graphs provide...

Elementary Proofs of Some Results on Representations of P-groups

A result of Roquette 3] states that if D is an absolutely irreducible representation of a p-group G over the eld of complex numbers, then D can be realized in K((g) j g 2 G), where is the character of D and K = Q or K = Q(i) according to whether p 6 = 2 or p = 2. Based on Baum and Clausen's 1] algorithm for computing the irreducible representations of supersolvable groups, we give an elementary...

Elementary abelian p - groups of rank 2 p + 3 are not CI - groups

For every prime p > 2 we exhibit a Cayley graph on Z2p+3 p which is not a CI-graph. This proves that an elementary abelian p-group of rank greater than or equal to 2p+ 3 is not a CI-group. The proof is elementary and uses only multivariate polynomials and basic tools of linear algebra. Moreover, we apply our technique to give a uniform explanation for the recent works of Muzychuk and Spiga conc...

Proofs in Elementary Analysis

Quillen’s Elementary Proofs

Recall 1.1. The construction of the Steenrod squares and powers proceeds in the following steps 1. For a space X, the p-th cartesian power Xp is acted on cyclically by Z/p and it fits into a fibration Xp → EZ/p×Z/p Xp → BZ/p 2. If x ∈ H∗(X, Z/p), one defines u × · · · × u ∈ H∗(Xp, Z/p) and extends this class to ũ ∈ H(EZ/p×Z/p Xp; Z/p) by a construction on the chain level. 3. ũ is then pulled ba...