Groups Definition and 867 Threads

Homework Statement Let G be a group, and let H be a normal subgroup of G. Must show that every subgroup K' of the factor group G/H has the form K'=K/H, where K is a subgroup of G that contains H. Homework Equations I don't see how to get started. The Attempt at a Solution I wrote...

Homework Statement Prove that the group of order 175 is abelian. Homework Equations The Attempt at a Solution |G|=175=527. Using the Sylow theorems it can be determined that G has only one Sylow 2-subgroup of order 25 called it H and only one Sylow 7-subgroups called it K. Thus...

[SIZE="5"][FONT="Comic Sans MS"][FONT="Courier New"]Let G be a Lie group. Show that there exists a unique affine connection such that \nabla X=0 for all left invariant vector fields. Show that this connection is torsion free iff the Lie algebra is Abelian.

Hello everyone. I need someone to explain a concept to me. I'm confused about how a type of particle can be a representation of a lie group. For example, I read that particles with half-integer spin j are a representation of the group SU(2), or that particles with charge q are a...

Homework Statement In the group <Z\stackrel{X}{13}> of nonzero classes modulo 13 under multiplication, find the subgroup generated by \overline{3} and \overline{10}Homework Equations The Attempt at a Solution Doesnt 3 generate {3,6,9,12} and 10 generate {2,5,10}?

V is a vectoric space. W_1,W_2\subseteq V\\ W_1\nsubseteq W_2\\ W_2\nsubseteq W_1\\ prove that W_1 \cup W_2 is not a vectoric subspace of V. i don't ave the shread of idea on how to tackle it i only know to prove that some stuff is subspace but constant mutiplication and by sum of two...

Homework Statement Recall that given a group G, we defined A(G) to be the set of all isomorphisms from G to itself; you proved that A(G) is a group under composition. (a) Prove that A(Zn) is isomorphic to Zn/{0} (b) Prove that A(Z) is isomorphic to Z2 Homework Equations The Attempt at a Solution

Homework Statement (a) How many distinct cyclic subgroups of D6 are there? Write them all down explicitly. (Here, D6 is the dihedral group of order 12, i.e. it is the group of symmetries of the regular hexagon.) (b) Exhibit a proper subgroup of D6 which is not cyclic. Homework...

Hello, I wonder if somebody could point me to a book (preferably), or paper, link, etc. which explores the relations between number theory and group theory. For example, I am (more or less) following Burton's "Elementary Number Theory" and there is no mention of groups. I also have the...

I was trying to prove the statement "If (2^n)-1 is prime then n is prime". I've already seen the proof using factorisation of the difference of integers and getting a contradiction, but I was trying to use groups instead. I was wondering if it's possible, since I keep getting stuck. So far I've...

Homework Statement suppose that G is a group in which every non-identity element has order two. show that G is commutative. Homework Equations The Attempt at a Solution IS THIS CORRECT? ab=a[(ab)^2]b=(a^2)(ba)(b^2)=ba

Homework Statement suppose that G is a group in which every non-identity element has order two. Show that G is commutative. Homework Equations The Attempt at a Solution Is my answer correct? Suppose that a,b and ab all have order two. we will show that a and b commute. By...

Homework Statement Suppose that G is a group in which every non-identity element has order two. show that G is commutative. Homework Equations The Attempt at a Solution DOES THIS ANSWER THE QUESTION?: Notice first that x2 = 1 is equivalent to x = x−1. Since every element of G...

I need help here: Suppose that G is a group in which every non-identity element has order two. Show that G is commutative. Also, Consider Zn = {0,1,...,n-1} a. show that an element k is a generator of Zn if and only if k and n are relatively prime. b. Is every...

I need help here: Suppose that G is a group in which every non-identity element has order two. Show that G is commutative. Also, Consider Zn = {0,1,...,n-1} a. show that an element k is a generator of Zn if and only if k and n are relatively prime. b. Is every...

Given a group G. J = {\phi: G -> G: \phi is an isormophism}. Prove J is a group (not a subgroup!). Well we know the operation is function composition. To demonstrate J a group we need to satisfy four properties: (i) Identity: (I'm not sure what to do with this) (ii) Inverses: Suppose a...

Hello, I am quite new here, as my number of posts might indicate. Thus I am not really sure whether or not this question should be posted here or somewhere else. It is not a homework, but neither is it a question that could not be a homework. However, here we go. I have, during a course in...

Hi, I am trying to compute homology groups of a space while some of the homotopy groups are known..what is the best way to do that. I hope that you can help. Thanks, Sandra

I'm studying the reaction mechanisms for carboxylic acid and its derivatives and here it says whether a compound with a C=O bond undergoes nucleophilic addition (as in aldehydes and ketones) or nucleophilic acyl substitution depends on the relative basicities of the substituent group. For...

I think I understand why a vector field must have a unique set of integral curves, but I don't see why they must define a one-parameter group of diffeomorphisms. Let X be a vector field on a manifold M, and p a point in M. A smooth curve C through p is said to be an integral curve of X if...

Can anyone expand on the relationship between Lie groups, Lie algebras, exponential maps and unitary operators in QM? I've been reading lately about Lie groups and exponential maps, and now I'm trying to tie it all together relating it back to QM. I guess I'm trying to make sense of how Lie...

Homework Statement Show that Z/mZ X Z/nZ isomorphic to Z/mnZ iff m and n are relatively prime. (Using first isomorphism theorem) Homework Equations The Attempt at a Solution Okay, first I want to construct a hom f : Z/mZ X Z/nZ ---> Z/mnZ I have f(1,0).m = 0(mod mn) =...

Hi, I have a question related to Group Theory and its interpretation from a social point of view. if we suppose, that a group of Humans can be considered as an algebraic structure : a group (G,◦) with a set of elements and a set of axioms like closure, associativity, identity and...

Hi, I would like to know if you guys know a good book on sets groups and relations, preferably with lots of exercises. I believe that I am on a beginner level, but I already know all basic concepts, so the text is not that important. It would be even better if it is available online hehe! Thks!

If A is a group and B is a subgroup of A, what is A/B? I don't need a definition really, just the term or name for the quantity A/B. I can look it up myself from there. Also, if anyone has any suggestions or links that make it easy to look up math concepts by notation rather than looking...

I am trying to do some self-study and plow through Ballentine's book on Quantum Mechanics. I thought I was following the majority of it until I got to Sec. 3.3, in particular the derivation of the multiples of identity for the commutators of these generators. For example, the commutator of...

1. The exponential map is a map from the lie algebra to a matrix representation of the group. For abelian groups, the group operation of matrix multiplication for the matrix rep clearly corresponds to the operation of addition in the lie algebra: \sum_a \Lambda_a t_a \rightarrow exp(\sum_a...

Homework Statement This is an exercise from Jacobson Algebra I, which has me stumped. Let G = G1 x G2 be a group, where G1 and G2 are simple groups. Prove that every proper normal subgroup K of G is isomorphic to G1 or G2. Homework Equations The Attempt at a Solution Certainly...

Hi. Need help with following problem: Let R=(V,E) a regular graph with degree at least 1 and odd number of vertices. Let C=Aut(R) the transitive action on the set E of R. Prove C also transitive action on the set V of R. Anyone got any idea/tips? Thanks!

I have a few questions: 1) The tensor product of two matrices is define by A \otimes B =\left( {\begin{array}{cc} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \\ \end{array} } \right) for the 2x2 case with obvious generalisation to higher dimensions. The tensor product of two...

Homework Statement How do i go about proving that a group is cyclic? Homework Equations The Attempt at a Solution

Suppose you had the following: (A,*) and (B,\nabla) So to prove associativity, since I know that both A and B are groups, their direct product will be a group. Could I do the following ai , bi \in A,B [(a1,b1)(a2,b2)](a3,b3)=(a1,b1)[(a2,b2)(a3,b3)] Since A and B are groups, I...

Hi, Just wondering who the big players in QG now, I know obviously Perimeter is huge, but what other research groups are there out there? Thanks

Homework Statement Count and describe the different isomorphism classes of abelian groups of order 1800. I don't need to list the group individually, but I need to give some sort of justification.Homework Equations The Attempt at a Solution I'm using the theorem to classify finitely generated...

Homework Statement Find a set of generators for a p-Sylow subgroup K of Sp2 . Find the order of K and determine whether it is normal in Sp2 and if it is abelian. Homework Equations The Attempt at a Solution So far I have that the order of Sp2 is p2!. So p2 is the highest power of...

Homework Statement Thus far in my studying I've been able to at least have a sense of where to start solving the problems... until now. Find the order of the subgroup of the multiplicative group G of 4x4 matrice generated by: | 0 1 0 0 | | 0 0 0 1 | | 0 0 1 0 | | 1 0 0 0 |...

Homework Statement Is there a good method for finding automorphism groups? I am currently working on finding them for D4 and D5. Homework Equations The Attempt at a Solution I've only really looked hard at D4 and the only one I've found is the identity. I know you have to send...

Homework Statement If G is a finite abelian group that has one subgroup of order d for every divisor d of the order of G. Prove that G is cyclic. Homework Equations The Attempt at a Solution

Homework Statement An abelian group has order pn (where p is a prime) and contains p-1 elements of order p. Prove that this group is cyclic. Homework Equations The Attempt at a Solution I know I should use the theorem for classifying finite abelian groups, which I understand, and...

Homework Statement Let G be a finite group and let p >= 3 be a prime such that p | |G|. Prove that the group ring ZpG is not a domain. Hint: Think about the value of (g − 1)p in ZpG where g in G and where 1 = e in G is the identity element of G. The Attempt at a Solution G is a...

Homework Statement Let p and q both be prime numbers and p > q. Classify groups of order p2q if p is not congruent to +1 or -1 mod q. Homework Equations The Attempt at a Solution It is clear that the Sylow theorems would be the things to use here. So I guess this says that the...

Prove R/Q has no element of finite order other than the identity. First of all, I have trouble visualizing what R/Q is. But I do know that afterwards you can try to raise an element in R/Q to a power to get to 0, but there will not be a finite number that will be able to do so except zero...

Find, up to isomorphism, all possible quotient groups of D6 and D9, the dihedral group of 12 and 18 elements. First of all, I don't understand the question by what they mean about "up to isomorphism." Does this mean by using the First Isomorphism Theorem? Also does this question imply that...

Why is Z mod 2 x Z mod 3 isomorphic to Z mod 6 but Z mod 2 x Z mod 2 not isomorphic to Z mod 4?

Homework Statement If G is a finite cyclic group of order n, what is the group Aut(G)? Aut(Aut(G))? Homework Equations The Attempt at a Solution Aut(G) is given by the automorphisms that send a generator to a power k < n where (k,n) = 1 with order p(n) where p is Euler's...

I'm looking at the space group #55, Pbam. In the top of the file (see below) it has listed: Pbam D^9_2h mmm Orthorhombic Is this saying that Pbam is consistent with point group mmm? It does not have three mirrors, it has two glide plane and one mirror. If I look at the...

Homework Statement I would like to find the number of distinct elements in S17 that are made up of two 4-cycles and three 3-cycles. Homework Equations The Attempt at a Solution This seems like a very simple question but since the group is so huge it's hard to figure out. I have...

Homework Statement In Z4 x Z4, find two subgroups H and K of order 4 such that H is not isomorphic to K, but (Z4 x Z4)/H isomorphic (Z4 x Z4)/K Homework Equations The Attempt at a Solution I know (Z4 x Z4) has twelve elements (0,0), (1,0), (2,0), (3,0), etc. I can generate subgroups of...

Alternating groups apply to all even permutations in Sn for n > 2. Since n = 2 is inclusive, what got me wondering is that for such a case there are only 2 elements in S (say w and x); wouldn't that mean that the only transposition permutation would be (w x), which is an odd permutation?

To be sure of things, groups do not necessarily have to have only one operation; they may have more, but there is one and only one operation they are identified with. Am I right?