The Unapologetic Mathematician

A subgroup is pretty straightforward. It’s a little group living inside a bigger group. If you’ve got a group and some collection of elements of so that is a group using the same composition as , then is a subgroup. To be more explicit, you need that

• If and are in then is in .
• If is in then is in .
• The identity of is in . [added at the suggestion of Toby Bartels]

We say that a subgroup is “closed” under composition and inverse, meaning that if we start with elements of and take compositions and inverses we never leave the subgroup.

The collection of all even integers is a subgroup of the group of all integers (with addition as the operation).
The subset is a subgroup of the group .
Every group has two “trivial” subgroups: the whole group itself, and the subgroup consisting of just the identity element.
There are two ways of getting subgroups that I want to spend a bit more time on: “images” and “kernels”.
Read more »

The New York Time reports that Princeton is closing its ESP research group: the Princeton Engineering Anomalies Research laboratory.

So, what does the Pyrus-in-chief have to say?

“For 28 years, we’ve done what we wanted to do, and there’s no reason to stay and generate more of the same data,” said the laboratory’s founder, Robert G. Jahn, 76. “If people don’t believe us after all the results we’ve produced, then they never will.”

In other words, “you can’t fire me, I quit”.

And the voice of (orthodox) science?

“It’s been an embarrassment to science, and I think an embarrassment for Princeton,” said Robert L. Park, a University of Maryland physicist. “Science has a substantial amount of credibility, but this is the kind of thing that squanders it.“

(emphasis added)
But Wait There’s More!

“We know people have ideas beyond the mainstream,” said the sociologist Harriet Zuckerman, “but if they want funds for research they have to go through peer review, and the system is going to be very skeptical of ideas that are inconsistent with what is already known.”

(ditto)

Some things do manage to make me happy in the news.