Wikipedia:Reference desk/Archives/Mathematics/2009 October 24

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October 24

Groups of order 112

How can I show that a group of order 112 cannot be simple? It may have 7 Sylow-2 subgroups which may be non cyclic and also may have 8 Sylow-7 subgroups. The Sylow-7 subgroups will have trivial intersections but the same cannot be said about the Sylow 2-subgroups? Thanks.-Shahab (talk) 05:55, 24 October 2009 (UTC)

If the normalizer mod the centralizer of every 2-subgroup is a 2-group, then the group is 2-nilpotent, and the Sylow 7-subgroup is normal. Assuming the group is not 2-nilpotent, it must have a 2-subgroup whose automorphism group has an element of order 7. However, the only 2-groups of order dividing 16 with an automorphism of order 7 are the elementary abelian groups of order 8 and 16. However, a subgroup of order 8 or 16 is normal in a group of order 16, and since it is also normalized by the element of order 7, it is normal in the whole group. Since the automorphism group of an elementary abelian group of order 8 does not contain a subgroup isomorphic to the dihedral group of order 14, the Sylow 2-subgroup itself is normal. In other words, every group of order 112 is either 2-nilpotent or 7-nilpotent (or both). JackSchmidt (talk) 07:17, 24 October 2009 (UTC)

Another approach: If G is not simple it has 7 Sylow 2-subgroups. 7 is not congruent to 1 mod 4 so two of these subgroups, say P and Q have an intersection of order 8. The normalizer of the intersection contains both P and Q which implies it must be G and so the intersection is normal. This is all assuming you're not allowed to use Burnside p^aq^b.--RDBury (talk) 18:54, 25 October 2009 (UTC)

Subset of an intersection

How to prove $A\subseteq B\cap C\Leftrightarrow A\subseteq B\land A\subseteq C$? --88.78.236.7 (talk) 12:01, 24 October 2009 (UTC)

This question has the appearance of a homework problem. You would want to prove it directly from the definitions of "$A\subseteq B\cap C$", "$A\subseteq B$", and "$A\subseteq C$". — Carl (CBM · talk) 12:07, 24 October 2009 (UTC)

HInt: To show $\Leftrightarrow$, individually show $\Leftarrow$ and $\Rightarrow$. I think this problem is one of the very early examples in Hopcroft/Motwani/Ullman. --Stephan Schulz (talk) 09:00, 25 October 2009 (UTC)

Grades formula

It sounds like a homework assignment, but I'm the teacher, not the student. I've told the class that 'weekly assignments' will count towards 10% of their grade, that 'contribution to seminar discussion, and participation' counts for 30%, and that the 'final paper' counts 60%. Grades will be mainly between 6 (low) and 10 (high), though mainly between 7-9. What formula can I use to calculate an overall grade between 6-10, if a student earns, for example, an '7' on weekly assignments at 10%, a '9' on classroom participation at 30%, and an '8' for the paper and 60%. Thanks if you can make it simple for me. —Preceding unsigned comment added by 80.101.134.43 (talk) 15:26, 24 October 2009 (UTC)

The usual way to do this is to add the grades, each multiplied by its weight (how much it counts towards). Note that $1\%={\frac {1}{100}}$ so $10\%=0.1$, etc. So the final grade will be $0.1\cdot 7+0.3\cdot 9+0.6\cdot 8=8.2$, and you can round it to 8 if necessary.

See also weighted mean. -- Meni Rosenfeld (talk) 17:58, 24 October 2009 (UTC)

A simple spreadsheet will do the calculations for you. Dbfirs 18:08, 24 October 2009 (UTC)

I've always wondered - how do you standardise the scoring? How do you decide what is worth an 8 rather than a 7? How do you ensure that this marking is consistant with the marking another teacher would give of the same work? 89.242.151.212 (talk) 22:24, 24 October 2009 (UTC)

You get other teachers to mark a selection of it and check they agree with you (you can also make sure the distribution of marks doesn't change too much from year to year - if you find you've given out twice as many 10's this year than last then you probably need to go back and change some of them). That may not happen for everything but work that contributes to a university degree, for example, needs to be moderated to ensure consistent standards between universities. At least, that's the way it works in the UK, other countries may have other systems. It doesn't work perfectly, though - UK universities tend to moderate the work of and get their work moderated by universities of a similar standard, which means a 1st from a university between 5th and 15th in the country, say, will be about the same as a 1st from another university in that range, but it can be completely different to a 1st from one of the top unis or one of the "New Universities". --Tango (talk) 22:40, 24 October 2009 (UTC)

Usually, I think that it is undesirable to score work on a large scale (1-10 is reasonable, but 1-30 is not). The level of difficulty in standardizing scores depends on the subject, as well. For instance, in the SAT writing section, two different people will mark one's essay (on a scale of 1-6) and if the two marks differ by more than a point, a third marker will resolve the conflict. This ensures that scores will be more standard in nature. With homework assignments (at university, of course), it is sometimes preferred to give a homework assignment either "no credit", "half credit" or "full credit" rather than on a scale of 1-10. This ensures that there will not be one point differences between good students in terms of homework assignments; rather, exams will discriminate the hard-working students. --PST 00:08, 25 October 2009 (UTC)

On the other hand, I should add that some unprofessional markers give marks such as 7.2/10 or 7.3/10, and then discriminate between students, which I think is quite silly (such marks are, of course, randomly decided). --PST 00:13, 25 October 2009 (UTC)

In my experience, I have found that my instructors give an amount of credit that is determined by a rubric. For example, a rubric for a mathematics assignment might dictate that each question is worth one point. If there were ten questions on the assignment and eight were answered, the instructor would calculate the grade as follows: $8\cdot 1=8$ The total credit for the assignment would be 8/10 or 80%. For an assignment that involves writing an essay, instructors might have a more complex rubric, assigning specific point values for formatting, organization, spelling and mechanics, and so forth. —Dromioofephesus (talk) 02:01, 25 October 2009 (UTC)

Tango - academics submit suggestions as to who might be a suitable external examiner for a programme. In fields where there are many possibilities, this is likely to be someone they know personally and believe is likely to accept the role. They may or may not be from a university perceived as being of a similar standard. For example, I work at a red brick university, and our department has externals from other redbricks, 60s universities, new universities and the Open University. Perhaps that isn't the case everywhere, but external's reports are becoming increasingly visible, and it will start to reflect badly on departments if they only source externals from very similar departments. Warofdreams talk 12:49, 26 October 2009 (UTC)

Maybe things are improving, but fair comparisons between widely different universities can't be taking place at the moment - if they were then there would be a strong correlation between average A-level results of entrants and percentage of firsts handed out (since there is a strong correlation between A-levels and degree classifications for individuals within one uni), and there isn't. --Tango (talk) 19:22, 26 October 2009 (UTC)

I can't believe such an elaborate rigmoral is gone through to mark student essays - you just get someone subjectively choosing what they think the right mark is. Similarly for school marking. 89.240.47.104 (talk) 23:37, 26 October 2009 (UTC)

If the essay doesn't count towards the degree, then sure, but you need to moderate the marking for work that counts otherwise the degree is completely meaningless. --Tango (talk) 23:42, 26 October 2009 (UTC)

Unfortunately, people tend to have extremely poor judgements these days; especially when it comes to marking essays, and especially in low-profile instituitions. Furthermore, marking essays requires experience, and several points of view. If more than one marker marks an essay, it is more likely that the marks will be based on strong writing, rather than on the point of view the essay develops. --PST 07:01, 28 October 2009 (UTC)

Worse judgement than in the past? It would be hard thing to demonstrate, but I've heard some horror stories from a few decades ago. Warofdreams talk 20:44, 28 October 2009 (UTC)

I have frequently had to assess work myself, and all the time its been complete guesswork. I was (now doing something else) supposed to put them in various grades, but there was never any guidance as to what was required for each grade. People overestimate their abilities to judge things objectively - really we are at the mercy of various biases (see List of cognitive biases ) and halo effects. Read books about Human judgement to find out about this. 92.29.91.83 (talk) 20:13, 28 October 2009 (UTC)

It's a worry to hear that you were never given any guidance. But a feeling for which pieces of work fulfil a brief or answer a question better does tend to develop over time. While, no doubt, different markers look for slightly different things, where I work, dissertations and postgraduate work is routinely double marked and the marks then compared; a difference of 5% is common, a difference of more than 10% very unusual. Warofdreams talk 20:44, 28 October 2009 (UTC)

Equalities of means

As far as I recall H^2 = AG where H is the harmonic mean, A is the artithmetic mean, G is the geometric mean and "^2" means squared. Does this only apply to the averages of two numbers or to the averages of any number of numbers please? (If it is not "HAG" then it is G^2=AH ). 89.242.151.212 (talk) 22:18, 24 October 2009 (UTC)

it's G² = AH, but it only works for two numbers. You can see that it fails in the more general case since G² is not guaranteed to be rational, but AH always is. Rckrone (talk) 02:18, 25 October 2009 (UTC)

Presumably you meant to say that G² is guaranteed to be rational for two terms when the terms are rational? --COVIZAPIBETEFOKY (talk) 13:40, 25 October 2009 (UTC)

Yeah sorry, you're right. Rckrone (talk) 15:56, 25 October 2009 (UTC)

Would using the formula for averages of more than 2 numbers be at approximately correct? 84.13.180.244 (talk) 00:12, 28 October 2009 (UTC)

Yes, if the numbers are close enough to one another. The error is cubic in the deviations. -- Meni Rosenfeld (talk) 21:04, 28 October 2009 (UTC)