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Cheryl Elisabeth Praeger, AM (born 7 September 1948, Toowoomba, Queensland) is an Australian mathematician. Praeger received BSc (1969) and MSc degrees from the University of Queensland (1974), and doctorate from the University of Oxford in 1973 under direction of Peter M. Neumann. She has published widely and has advised 20 Ph.D. students (as of May 2008). She is currently a professor of mathematics at the University of Western Australia. She is best known for her works in group theory, algebraic graph theory and combinatorial designs.

She has Erdős number 2

Education
Praeger completed her high school education at Brisbane Girls Grammar School. After graduating high school, Praeger went to the government vocational guidance section to inquire about how she could further study mathematics. The vocational guidance officer she spoke with tried to discourage her from studying mathematics further, suggesting she become a teacher or a nurse because two other girls who came to him wanting to study math weren't able to pass their courses. He reluctantly showed her an engineering course description, but she felt it didn't have enough mathematics. So she left without getting much information that day, but did continue on to receive her Bachelor's and Master's degrees from University of Queensland.

Having met several women on the mathematics staff during her undergraduate studies, the prospect of becoming a mathematician didn't seem strange to her. During her first and second years she did honors studies in mathematics and physics, choosing to continue in mathematics after her second year. After completing her education at University of Queensland she was offered a research scholarship at ANU but chose instead to take the Commonwealth Scholarship to the University of Oxford and attended St Anne's College. At that point she knew she wanted to study alegbra.

After earning her doctorate in 1973 she had a research fellowship at ANU. She had her first opportunity at teaching regular classes at the University of Virginia during the semester she worked there. Afterwards she returned to ANU where she met her future husband, John Henstridge, who was studying Statistics. She was later offered a short-term position at the University of Western Australia, which turned into a long term position, where she currently works today.

She has taught in the Mathematics and Statistics program at UWA and was Head of the Department of Mathematics 1992-1994, inaugural Dean of Postgraduate Research Studies 1996-1998, Chair Promotions and Tenure Committee 2000-2004, Deputy Dean of the Faculty of Engineering Computing and Mathematics 2003-2006, and ARC Professorial Fellow 2007.

Awards, Honors and Memberships
Praeger is a Fellow of the Australian Academy of Science, former president of the Australian Mathematical Society (1992-1994), and was appointed as a member of the Order of Australia in 1999 for her service to mathematics in Australia, especially through research and professional associations. In 2003 she received the Centenary Medal of the Australian Government. She was awarded an honorary DSc by the Prince of Songkla University in 1993 and by the Université Libre de Bruxelles in 2005. In 2007 she was awarded an Australian Research Council Federation Fellowship grant. She was named WA Scientist of the Year in 2009.

Praeger has also held memberships with the Combinatorial Mathematics Society of Australasia, Institute of Combinatorics and its Applications, Australian Mathematics Trust, American Mathematical Society, and the London Mathematical Society. Her past affiliations have not been limited to academia. She has also been a member of the Curriculum Development Center of the Commonwealth Schools Commission, Science Advisory Committee, WISET Advisory Committee to the Federal Minister for Science on participation of women in Science, Engineering, and Technology, UWA Academy of Young Mathematicians Lectures, the Western Australian School Mathematics Enrichment Course Tutor, and Data Analysis Australia Pty Ltd. She has also served on the Australian Federation of University Women (Western Australian Branch) and the Nedlands Primary School Council.

Marriage, Family and Other Interests
In August 1975 she married John Henstridge in Brisbane. They have two children, James (1979) and Tim (1982).

In addition to holding a doctorate in mathematics, she also holds an AMusA in piano performance and is a member of the University of Western Australia Collegium Musicum. She has been a member of the Uniting Church in Australia, Nedlands Parish since 1977, functioned as an elder from 1981-1987, and as an organist since 1985. She lists keyboard music among her stronger interests along with sailing, hiking, and cycling.

Today Praeger also promotes the involvement of women in mathematics by encouraging girls in primary and secondary schools with lectures, workshops, conferences and through Family Maths Program Australia (FAMPA), which she was key in implementing in local primary schools.

Work
Praeger's key research is in Group Theory and Combinatorics, including Analysis of algorithms and complexity, Discrete Mathematics and Geometry. She was first published in 1970 while still an undergraduate. As of April 2013, she has 328 publications total.

She has co-authored several papers on symmetric graphs and distance-transitive graphs and with Tony Gardiner.

With Jan Saxl and Martin Liebeck, she has co-authored papers on many topics including: permutation groups, primitive permutation groups, simple groups, and almost simple groups. Together they co-authored "On the O'Nan Scott Theorem for primitive permutation groups" it pertains to the classification of finite simple groups, namely the classification of finite primitive permutation groups. The paper contains a complete self-contained proof of the theorem.

She also co-authored "On the Sims' conjecture and distance transitive graphs" with Peter Cameron, G.M. Seitz and Jan Saxl. (cite it) It has important applications in (insert appropriate fields)

The O'Nan-Scott Theorem
The O'Nan-Scott Theorem is one of the most influential theorems of permutation group theory, the classification of finite simple groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric group. It appeared as an appendix to a paper by Leonard Scott written for The Santa Cruz Conference on Finite Groups in 1979, with a footnote that Michael O'Nan had independently proved the same result.

The theorem states that that a maximal subgroup of the symmetric group Sym(Ω), where |Ω| = n is one of the following:
 * 1) Sk × Sn-k the stabilizer of a k-set (that is, intransitive)
 * 2) SawrSb with n = ab, the stabilizer of a partition into b parts of size a (that is imprimitive)
 * 3) primitive (that is, preserves no nontrivial partition) and of one of the following types:
 * AGL(d,p)
 * SlwrSk, the stabilizer of the product structure Ω = Δk
 * a group of diagonal type
 * an almost simple group

In their paper, "On the O'Nan Scott Theorem for primitive permutation groups," M.W. Liebeck, Cheryl Praeger and Jan Saxl give a complete self contained proof of the theorem. In addition to the proof, they recognized that real power in the O'Nan-Scott theorem is in the ability to split the finite primitive groups into various types.

Given a transitive permutation group Gwith nontrivial normal subgroup N, the orbits of N form a system of of imprimitivity for G

The Eight O'Nan-Scott Types
The eight O'Nan-Scott types are as follows:

HA (holomorph of an abelian group): These are the primitive groups which are subgroups of the affine general linear group AGL(d,p), for some prime p and positive integer d ≥ 1. For such a group G to be primitive, it must contain the subgroup of all translations, and the stabilizer G0 in G of the zero vector must be an irreducible subgroup of GL(d,p). Primitive groups of type HA are characterized by having a unique minimal normal subgroup which is elementary abelian and acts regularly.

HS (holomorph of a simple group): Let T be a finite nonabelian simple group. Then M = T×T acts on Ω=T by t(t1,t2) = t1-1tt2. Now M has two minimal normal subgroups N1, N2, each isomorphic to T and each acts regularly on Ω, one by right multiplication and one by left multiplication. The action of M is primitive and if we take α = 1T we have Mα = {(t,t)|t ∈ T}, which includes Inn(T) on Ω. In fact any automorphism of T will act on Ω. A primitive group of type HS is then any group G such that M ≅ T.Inn(T)≤G≤T.Aut(T). All such groups have N1 and N2 as minimal normal subgroups.

HC (holomorph of a compound group): Let T be a nonabelian simple group and let N1≅ N2≅ Tk for some integer k ≥ 2. Let Ω = Tk. Then M = N1 × N2acts transitively on Ω via x(n1,n2) = n1-1xn2 for all x ∈ Ω, n1 ∈ N1, n2 ∈ N2. As in the HS case, we have M ≅ Tk.Inn(Tk) and any automorphism of Tk also acts on Ω. A primitive group of type HC is a group G such that M≤G≤ Tk.Aut(Tk)and G induces a subgroup of Aut(Tk) = Aut(T)wrSk which acts transitively on the set of k simple direct factors of Tk. Any such G has two minimal normal subgroups, each isomorphic to Tk and regular.

A group of type HC preserves a product structure Ω = Δk where Δ = T and G≤ HwrSk where H is a primitive group of type HS on Δ.

TW (twisted wreath):Here G has a unique minimal normal subgroup N and N ≅ Tk for some finite nonabelian simple group T and N acts regularly on Ω. Such groups can be constructed as twisted wreath products and hence the label TW. The conditions required to get primitivity imply that k≥ 6 so the smallest degree of such a primitive group is 606.

AS (almost simple): Here G is a group lying between T and Aut(T ), that is, G is an almost simple group and so the name. We are not told anything about what the action is, other than that it is primitive. Analysis of this type requires knowing about the possible primitive actions of almost simple groups, which is equivalent to knowing the maximal subgroups of almost simple groups.

SD (simple diagonal): Let N = Tk for some nonabelian simple group T and integer k ≥ 2 and let H = {(t,...,t)| t ∈ T} ≤ N. Then N acts on the set Ω of right cosets of H in N by right multiplication. We can take {(t1,...,tk-1, 1)| ti ∈ T}to be a set of coset representatives for H in N and so we can identify Ω with Tk-1. Now (s1,...,sk) ∈ N takes the coset with representative (t1,...,tk-1, 1) to the coset H(t1s1,...,tk-1sk-1, sk) = H(sk-1tks1,...,sk-1tk-1sk-1, 1)The group Sk induces automorphisms of N by permuting the entries and fixes the subgroup H and so acts on the set Ω. Also, note that H acts on Ω by inducing Inn(T) and in fact any automorphism σ of T acts on Ω by taking the coset with representative (t1,...,tk-1, 1)to the coset with representative (t1σ,...,tk-1σ, 1). Thus we get a group W = N.(Out(T) × Sk) ≤ Sym(Ω). A primitive group of type SD is a group G ≤ W such that N ◅ G and G induces a primitive subgroup of Sk on the k simple direct factors of N.

CD (compound diagonal): Here Ω = Δk and G ≤ HwrSk where H is a primitive group of type SD on Δ with minimal normal subgroup Tl. Moreover, N = Tkl is a minimal normal subgroup of G and G induces a transitive subgroup of Sk.

PA (product action): Here Ω = Δk and G ≤ HwrSk where H is a primitive almost simple group on with socle T. Thus G has a product action on Ω. Moreover, N = Tk ◅ G and G induces a transitive subgroup of Sk in its action on the k simple direct factors of N.

Some authors use different divisions of the types. The most common is to include types HS and SD together as a “diagonal type” and types HC, CD and PA together as a “product action type." Praeger later generalized the O’Nan-Scott Theorem to quasiprimitive groups.