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Topic: Sylow subgroup

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Normal subgroup

Dickcissel

Allegheny Front

	Third Sylow Theorem (Site not responding. Last check: 2007-10-10)
		The number of sylow subgroups divides m, and is equal to 1 mod p.
		The size of the orbit, which is the number of sylow subgroups, equals the index of the normalizer of h in g.
		The sylow subgroup j is fixed by h iff yj/y = j for all y in h, iff h is contained in the normalizer of j.
	www.mathreference.com /grp-act,sylow3.html (391 words)

	Sylow theorem : Sylow theorems
		The Sylow theorems of group theory form a partial converse to the theorem of Lagrange, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G.
		Let p be a prime number; then we define a Sylow p-subgroup of G to be a maximal p-subgroup of G (i.e., a subgroup which is a p-group, and which isn't a proper subgroup of any other p-subgroup of G).
		In fact, H (being in the center of G) is a normal subgroup of G.
	www.findword.org /sy/sylow-theorems.html (1647 words)

	Spring 1980 Algebra Prelim Solutions
		Sylow's theorem for the prime 7 shows that the number of Sylow 7-subgroups is congruent to 1 modulo 7 and divides 8, hence the number of Sylow 7-subgroups is 1 or 8.
		If there is one Sylow 7-subgroup, then this subgroup must be normal in G which contradicts the hypothesis that G is simple, consequently G has 8 Sylow 7-subgroups.
		Therefore G has exactly one Sylow 2-subgroup, and so this Sylow subgroup must be normal which contradicts the hypothesis that G is simple.
	www.math.vt.edu /people/linnell/Teaching/Algprelims/Spring80sol (625 words)

	Group Theory - Sylow Groups (Site not responding. Last check: 2007-10-10)
		All subgroups conjugate to a Sylow group are themselves Sylow groups.
		Sylow groups of order 3, thus the total number of elements is greater than 30, a contradiction.
		Its preimage under the natural map is a normal subgroup whose order is a multiple of 5, which we have previously shown to be a contradiction.
	rooster.stanford.edu /~ben/maths/group/sylow.php (449 words)

	Sylow theorem - Freepedia (Site not responding. Last check: 2007-10-10)
		In mathematics, especially group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G.
		Let p be a prime number; then we define a Sylow p-subgroup of G to be a maximal p-subgroup of G (i.e., a subgroup which is a p-group, and which is not a proper subgroup of any other p-subgroup of G).
		The problem of finding a Sylow subgroup of a given group is an important problem in computational group theory.
	en.freepedia.org /Sylow.html (1114 words)

	No Title (Site not responding. Last check: 2007-10-10)
		The subgroups of G/K are in 1-1 correspondence with the set of subgroups of G which contain K.
		(a) Since the number of 5-Sylow subgroups have to divide 30 and be equal 1 mod 5, it is 1 or 6.
		be the subgroup of G consisting of all elements of period 2, i.e.
	www.math.gatech.edu /~saugata/teaching/fall00/sol2/sol2.html (684 words)

	Classifying Finite Nilpotent Groups (Site not responding. Last check: 2007-10-10)
		If x belongs to two separate sylow subgroups, its order is a power of a prime p, and a power of some other prime q, which is impossible.
		Let h and k be two sylow subgroups and let d be the group generated by h and k.
		If j is a third sylow subgroup, it is disjoint from d, since everything in d has order divisible by p and q, and everything in j has order divisible by some other prime r.
	www.mathreference.com /grp-chain,fng.html (484 words)

	Subgroups
		That is, if H is a subgroup of the generic abelian group A, then, as a general rule, the structure of H is not computed at creation time.
		In this case, the list L would be empty since computing the subgroup's structure will be achieved by building the p-Sylow subgroups from random elements (of the subgroup).
		Also, if the group structure of A is already known/computed, and if the subgroup is defined in terms of a set of generators in L then the subgroup structure is computed at the time of creation.
	www.umich.edu /~gpcc/scs/magma/text263.htm (483 words)

	[No title]
		Using part(a) and part(b) We know that the size of the orbit of H is both a multiple of p and not a multiple of p.
		The source of this contradiction was the assumption that there is some K outside the orbit of H. This means that all of the p-sylow subgroups are in the orbit of H and consequently, all of the p-sylow subgroups are conjugate.
		But when K is a p-sylow subgroups there is a contradiction because the above elements form e a group with more than p elements.
	orion.math.iastate.edu /hentzel/class.301.03/Sep.26 (938 words)

	[No title]
		It is a subgroup of Co3 of index equal to 11178 = 2.* * 35.
		There are nine such subgroups up to conjugacy in S. Explicit gener* *ators are described in x3 via their restrictions to these nine centralizers.
		In the case of Kfiit was known that the cohomology is detected on* * the centralizers of the maximal elementary abelian subgroups and this aided the com* *putation.
	hopf.math.purdue.edu /Adem-Carlson-Karagueuzian-Milgram/hs.txt (7182 words)

	[No title]
		In a group of order $40$ the number of Sylow 5-subgroups must be 1 by Sylow's theorem, so it cannot be simple.
		If each pair of distinct Sylow 3-subgroups intersects in the trivial group then there would be 80 elements of order 3 or 9.
		Sylow's theorem also tells us that there are 21 subgroups of order 5 and therefore 84 elements of order 5 in $G$.
	www.bath.ac.uk /~masgcs/math0038/t8s.txt (1106 words)

	Hall subgroup (Site not responding. Last check: 2007-10-10)
		In mathematics, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index.
		A Hall subgroup of G is a subgroup whose order is a Hall divisor of the order of G.
		Any Sylow subgroup of a group is a Hall subgroup.
	www.worldhistory.com /wiki/H/Hall-subgroup.htm (423 words)

	Sylow p-Subgroups (Site not responding. Last check: 2007-10-10)
		A Sylow p-Subgroup of the group G is a subgroup of order p
		Sylow's Theorems : If G is a finite group and p is a prime number, then,
		A Sylow p-subgroup of order p will of course be a cyclic subgroup.
	hemsidor.torget.se /users/m/mauritz/math/alg/sylow.htm (146 words)

	Standard Subgroup Constructions
		Given groups H and K, both subgroups of the group G, construct the commutator subgroup of H and K in the group G. If K is a subgroup of H, then the group G may be omitted.
		Construct the centralizer of the permutation g in the group G; g and G must belong to a common symmetric group.
		Given a subgroup H of the permutation group G, construct the maximal normal subgroup of G that is contained in the subgroup H. H ^ G : GrpPerm, GrpPerm -> GrpPerm
	www.math.uiuc.edu /Software/magma/text244.html (360 words)

	Sylow's theorems
		(c) The number of Sylow p-subgroups of G divides the order of G and is equal to 1 modulo p.
		We are therefore assured the existence of a subgroup T in N
		The number of Sylow p-subgroups is the length of the orbit, which in turn divides the order of G.
	www.pitt.edu /~gmc/ch1/node6.html (521 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, §7.4 Solved Problems
		The number of Sylow 7-subgroups must be congruent to 1 mod 7 and must be a divisor of 24.
		Therefore the Sylow p-subgroups are precisely the cyclic subgroups of order p, each generated by a p-cycle.
		Therefore PQ is a subgroup, and it must be normal since its index is the smallest prime divisor of G, so we can apply the result in the previous problem.
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/soln74.html (840 words)

	The second and third Sylow Theorems
		This is the content of the second Sylow Theorem.
		We come now to the last of the three Sylow theorems.
		A few of their applications will be seen in the examples of the next section.
	web.usna.navy.mil /~wdj/tonybook/gpthry/node56.html (388 words)

	Useful Theorems
		Sylow's Theorem says there there is a subgroup
		If G is a finite group and H is a proper subgroup of G such that
		is a normal subgroup of G contained in H, and
	www-math.cudenver.edu /~rrosterm/simple/node2.html (315 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 7.4
		Show that a group of order 108 has a normal subgroup of order 9 or 27.
		Prove that if G is a group of order 56, then G has a normal Sylow 2-subgroup or a normal Sylow 7-subgroup.
		Prove that if G is a group of order 105, then G has a normal Sylow 5-subgroup and a normal Sylow 7-subgroup.
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/74.html (437 words)

	Subgroups
		Also, where the group structure of A is already known, or has been computed, and if the subgroup is defined in terms of a set of generators in L then the subgroup structure is computed at the time of creation.
		The next statements create two subgroups of GA_(qf) and determine their structure.
		The following statements create each of the p-Sylow subgroups of G and determine their structure.
	www.math.lsu.edu /magma/text347.htm (586 words)

	GAP Forum: Re: Memory Use
		classes of subgroups of a group G and g ranges over elements of G. have Gap the group G = SymmetricGroup(6), which is not too large (720).
		the Sylow subgroup of the representative itself is abelian, before you
		Sylow subgroups of the representatives are also copied.
	www-gap.dcs.st-and.ac.uk /oldsite/Forum/Schoener.1/Martin.1/Re__Memo.2/1.html (534 words)

	Homework 7, 22M:120 (Site not responding. Last check: 2007-10-10)
		(a) LET H BE A SUBGROUP OF A FINITE GROUP G. p-SUBGROUP OF G. INTERSECTION OF P' AND H IS A p-SYLOW SUBGROUP OF H. (b) ASSUME H NORMAL AND (G : H) PRIME TO p.
		LET N(H) DENOTE THE NORMALIZER OF A SUBGROUP H. LET P BE A p-SYLOW SUBGROUP OF G. To show that N(N(P)) is a subgoup of N(P), it might be helpful to first show that for all g in N(N(P)), gPg
		is a Sylow p-subgroup of N(P) and then to show that N(P) has only one Sylow p-subgroup.
	www.math.uiowa.edu /~fbleher/m120set7.html (265 words)

	GAP Manual: 59.10 Further Information (Site not responding. Last check: 2007-10-10)
		In general, the programs run most efficiently if the indices between successive terms in this sequence are as small as possible.
		itself, the sequence of subgroups must be strictly decreasing, and the last term must be equal to the Sylow subgroup stored as
		If that had been false, we could have replaced chr.sylow by a Sylow 2-subgroup of H2.
	www-gap.dcs.st-and.ac.uk /~gap/Gap3/Manual3/C059S010.htm (249 words)

	The world's top sylow theorem websites
		The Sylow theorems of group theory, named after Ludwig Sylow, form a partial converse to the theorem of Lagrange, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G.
		The following facts were first proposed and proven by Norwegian mathematician Ludwig Sylow in 1872, and published in Mathematische Annalen.
		Then for any element x which is not in the center of G, Z(G), we find that the centralizer of x, C(x), is a subgroup of G; and so will not be divisible by p
	dirs.org /wiki-article-tab.cfm/sylow_theorem (1389 words)

	Existence of Sylow subgroups; the first Sylow Theorem
		Existence of Sylow subgroups; the first Sylow Theorem
		We now proceed to the main result of this section, i.e., the first Sylow Theorem.
		On the basis of this theorem, we can now strengthen the result obtained in Theorem 10.1.2.
	web.usna.navy.mil /~wdj/tonybook/gpthry/node54.html (241 words)

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