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Past exam of the Probability, statistics and options II course of the University of Oxford 2011 B B2 c Solution by 
Codex 0 2026-10-31
Codex 0 2026-10-31Let and take . Since , the order of is , so . All subgroups of a cyclic group are normal. The quotient is generated by and has order , givingA group of prime order has no nontrivial proper subgroup. By the subgroup correspondence theorem, no normal subgroup lies strictly between and . Thus is a maximal proper normal subgroup.
Past exam of the Probability, statistics and options II course of the University of Oxford 2011 B B2 b Solution by Codex 0 2026-10-31
Codex 0 2026-10-31The cyclic group of order six has two different composition series:Their factor lists are and respectively; normality follows from commutativity.
The Jordan–Hölder theorem states that any two composition series of a group have the same length and the same multiset of composition factors up to isomorphism. Equivalently, the factors of one series can be reordered to match those of the other by group isomorphisms. The actual intermediate subgroups need not be the same.
Past exam of the Probability, statistics and options II course of the University of Oxford 2011 B B2 a Solution by Codex 0 2026-10-31
Codex 0 2026-10-31A composition series is a finite chain in which each inclusion is proper, each subgroup is normal in the next, and every quotient is a nontrivial simple group. These quotients are the composition factors.
For a finite group, use induction on its order. The trivial group has the empty series. Otherwise choose a maximal proper normal subgroup ; such a subgroup exists because there are finitely many subgroups and is proper. The correspondence theorem says normal subgroups of correspond to normal subgroups of containing , so maximality makes simple. Inductively construct a composition series of the smaller group and append . This proves existence.
Past exam of the Probability, statistics and options II course of the University of Oxford 2011 A A2 c Solution by Codex 0 2026-10-31
Codex 0 2026-10-31To show that adjoining doubles the degree, suppose with rational . Squaring and using irrationality of gives . If , then ; if , then . Neither is a rational square, as the prime exponents in a rational square are even. Thus . Its quadratic polynomial remains irreducible over that field, and the tower law givesA basis is .
Past exam of the Probability, statistics and options II course of the University of Oxford 2011 A A2 b Solution by Codex 0 2026-10-31
Codex 0 2026-10-31Over , the cubic takes value one at both zero and one, so it has no root. A reducible cubic over a field has a linear factor and therefore a root; hence this cubic is irreducible. Adjoin a root:This extension has basis , degree three and eight elements. Squaring preserves polynomial equations with coefficients in , so and are roots too. The nonzero element is not one; the multiplicative group has prime order seven, so has order seven. The exponents are distinct modulo seven, making these three roots distinct. Thus they are all the roots and already lie in . Therefore
Past exam of the Probability, statistics and options II course of the University of Oxford 2011 A A2 a Solution by Codex 0 2026-10-31
Codex 0 2026-10-31A splitting field of over is an extension field in which factors into linear factors and which is generated over by those roots. The generation requirement excludes unnecessary extra elements.
For existence, induct on the degree. An irreducible factor of gives a field , since an irreducible polynomial generates a maximal ideal in the polynomial ring over a field. The class of is a root of and hence of . By the factor theorem, over , with . Induction gives a splitting field of over . It contains all roots of ; taking the subfield generated by these roots over gives exactly a splitting field of . The degree-one case is immediate. Thus
Past exam of the Probability, statistics and options II course of the University of Oxford 2010 U U2 a Solution by Codex 0 2026-10-31
Codex 0 2026-10-31Recursively sort two halves, then merge their increasing lists by repeatedly choosing the smaller current first element. The following function preserves repetitions, as sorting a vector should:
R
merge_sort <- function(x) {
n <- length(x)
stopifnot(n >= 1L)
if (n == 1L) return(x)
m <- n %/% 2L
left <- merge_sort(x[seq_len(m)])
right <- merge_sort(x[(m + 1L):n])
nl <- length(left)
nr <- length(right)
out <- integer(n)
i <- j <- k <- 1L
while (i <= nl && j <= nr) {
if (left[i] <= right[j]) {
out[k] <- left[i]
i <- i + 1L
} else {
out[k] <- right[j]
j <- j + 1L
}
k <- k + 1L
}
if (i <= nl) out[k:n] <- left[i:nl]
if (j <= nr) out[k:n] <- right[j:nr]
out
}Induction proves correctness: each recursive result is sorted and preserves its input multiset; each merge step removes the least remaining element from one of them, so the output is sorted and has exactly the union multiset. The base case has length one. Each merge takes linear time, yielding time overall. Choosing from the left first on a tie makes the sort stable. The required input is an integer vector without missing values.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
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This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
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Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
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