Groups Definition and 906 Threads

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...

Hey trying to understand group theory a bit more. Need some help understanding how to spot which groups are isomorphic to what. First of all, but silly, but confirmation on what the groups D is dihedral right? What sort of other groups with i have to deal with. Like Z for integers? and U for...

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?

Hi, everyone: I have been looking for a while without success, for the definition of equivalence for unimodular quadratic forms defined on Abelian groups . I have found instead ,t the def. of equivalence in the more common case where the two forms Q,Q' are defined on vector...

Hi everyone! I would like to solve some questions: Classify up to isomorphism the four-sheeted normal coverings of a wedge of circles. describe them. i tried to to this and it is my understanding that such four sheeted normal coverings have four vertices and there are loops at each of...

If A and B are subsets of G, let A*B denote the set of all points a*b for a in A and b in B. Let A^(-1) denote the set of all points a^(-1), for a in A. a)A neighborhood V of the identity element e is said to be symmetric if V = V^(-1) . If U is a neighborhood of e, show there is a...

Homework Statement I am interested in proving that A5 has no normal subgroups except itself and {e}. The Attempt at a Solution Some proofs that I have seen use centralizers to do this, but since I haven't gone through that yet I think there should be some say to do it without them. My...

Let n be a positive integer and let m be a factor of 2n. Show that Dn (the dihedral group) contains a subgroup of order m. I'm not really sure where to start with this one. I know that Dn is generated by two types of rotations: flipping the n-gon over about an axis, and rotating it 2π/n...

Let An (the alternating group on n elements) consist of the set of all even permutations in Sn. Prove that An is indeed a subgroup of Sn and that it has index two in Sn and has order n!/2. First of all, I need clarification on the definition of an alternating group. My book wasn't really good...

Homework Statement Partition the following collection of groups into subcollections of isomorphic groups. a * superscript means all nonzero elements of the set. integers under addition S_{2} S_{6} integer_{6} integer_{2} real^{*} under multiplication real^{+} under multiplication...

Homework Statement Recall that Vm is the set of all invertibles in Z/m a) List the elements in V15 b) Find all the subgroups of V15 c) Is V15 cyclic? why? Homework Equations The Attempt at a Solution From my notes: a) V15 = {1, 2, 4, 7, 8, 11, 13, 14} b)...

Homework Statement Let H be the subgroup of GL(3, \mathbb{Z}_3) consisting of all matrices of the form \left[ \begin{array}{ccc} 1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1 \end{array} \right], where a,b,c \in \mathbb{Z}_3. I have to prove that Z(H) is isomorphic to \mathbb{Z}_3 and that H/Z(H) is...

Homework Statement A) If M is any monois, let M' denote the set of all nonempty subsets of M and define an operation on M' by XY = {xy | x in X, y in Y}. show that M' is a monoid, commutative if M is, and find the units. B) If ab=ba in a monoid M, prove that (ab)^n = a^nb^n for all n >=...

Is there a formula for determining the number of different groups up to isomorphism for a group of a given order?

Hello, What about gauging discrete groups ? (C, P, T (??), Flavour Groups, Fermionic number symmetry...)

1. The problem statement, all variables and given/known data Let a,b, be positive integers, and let d=gcd(a,b) and m=lcm(a,b). Show ZaXZb isomorphic to ZdXZm Homework Equations m=lcm(a,b) implies a|m, b|m and if a,b|c then m|c. d=gcd(a,b) implies d|a, d|b and if c|a and c|b then d|c...

Is it true that any group of order 30 has a normal (hence unique) Sylow-5 subgroup? I know that that the only possibilities for n(5) are 1 or 6. Now suppose there are 6 sylow 5 subgroups in G. This would yield (5-1)6=24 distinct elements of order 5 in G. Now there is only 30-24=6 elements left...

Homework Statement Let f : G → H be a homomorphism of Abelian groups. 1. Show that f (0) = 0. 2. Show that f (−x) = −f (x) for each x ∈ G. Homework Equations The Attempt at a Solution My background in topology / group theory is next to nothing. 1. Show that f(0) = 0. My attempt is as...

Do they exist? What are some examples? Are there any applications? What are some good books on the topic?

Let H be normal in G, H cyclic. Show any subgroup K of H is normal in G. I was thinking about using the fact that subgroups of cyclic groups are cyclic, and that subgroups of cyclic groups are (fully)Characteristic (is that true?). Then we would have K char in H and H normal in G. Hence K...

I have two questions, they aren't homework questions but I figured this would be the best place to post them (they are for studying for my exam). Homework Statement How many elements of S_6 have order 4? Do any elements have order greater than 7? Homework Equations S_6 is the...

I've taken a course in Linear Algebra, so I'm used to working with vector spaces. But now, I'm reading Griffith's Introduction to Elementary Particles, and it talks about groups having closure, an identity, an inverse, and being associative. With the exception of commutativity (unless the...