P (band)

P was a short-lived American alternative rock band formed in early 1993 by Butthole Surfers frontman Gibby Haynes (vocals), actor Johnny Depp (guitar/bass), actor Sal Jenco (percussion), and songwriter Bill Carter (guitar/bass).

History

The band performed their first show at the Austin Music Awards in 1993 and released their eponymous debut album on November 21, 1995 under Capitol Records. It was reissued on May 8, 2007 under Caroline Records. They often played the odd gig at The Viper Room, of which Depp used to be co-owner. One of these gigs was played on October 30, 1993, where the lineup included Flea of Red Hot Chili Peppers. Haynes, along with other members of the line-up that night, was a good friend of actor River Phoenix. While the band were in the middle of their song "Michael Stipe", which includes the lines "I'm glad I met old Michael Stipe, I didn't get to see his car. Him and River Phoenix were leaving on the road tomorrow" and "but we didn't have a part, not a piece of our heart, not Michael, River Phoenix or Flea or me," Phoenix (unbeknownst to the band at the time) was outside the venue having seizures on the sidewalk. Phoenix died in the early hours of October 31 of heart failure brought on by an overdose of cocaine and heroin.

Prüfer group

In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p^∞-group, Z(p^∞), for a prime number p is the unique p-group in which every element has p different p-th roots.

The Prüfer p-groups are countable abelian groups which are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups.

The groups are named after Heinz Prüfer, a German mathematician of the early 20th century.

Constructions of Z(p^∞)

The Prüfer p-group may be identified with the subgroup of the circle group, U(1), consisting of all pⁿ-th roots of unity as n ranges over all non-negative integers:

The group operation here is the multiplication of complex numbers.

Alternatively and equivalently, the Prüfer p-group may be defined as the Sylow p-subgroup of the quotient group Q/Z, consisting of those elements whose order is a power of p:

(where Z[1/p] denotes the group of all rational numbers whose denominator is a power of p, using addition of rational numbers as group operation).

P3 group

The P3 group, (formerly also P3 Ingenieurgesellschaft), is a global management consultancy with headquarters in Aachen, Germany. Based on the company’s data, approximately 2700 employees worked for the company in 2014, generating revenues of approximately 280 million euros. The P3 group draws its clients primarily from the automotive, energy, aerospace, and telecommunications industries.

Formation

P3 Ingenieurgesellschaft mbH is a spin-off company of the Fraunhofer Institute for Production Technology (IPT) and was founded in 1996 in Aachen by Thomas Prefi, Christoph Theis and Thomas Weingarten. The formation of the company was the result of the successful development of a quality management system for the Daimler-Benz AG.

The mentor and doctoral advisor of the three founders was Prof. Tilo Pfeifer, the head of production measurement technology and quality management at the RWTH Aachen and chairman of the scientific advisory committee of the German Society for Quality (DGQ).

Subsidiaries and international development