I've seen that there are different groups like $SU(2)$ and $SO(3)$ with isomorphic Lie algebra. I know that putting the elements of the linear algebra in the exponential function gives the group elements, in this case I don't understand how it's possible that two isomorphic linear algebras gives different groups  Does it mean that when I put two isomorphic group in the exponential function I can obtain two non isomorphic groups?
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I thought that groups with the same Lie Algebra are automatically equivalent, but there appear to be some exceptions to this?   What sort of exceptions are there and why?
110811
Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more explicit way or is this explicit enough?    Problem Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.  Solution Recall that isomorphism preserves order of elements and hence if there exists an isomorphism from $\phi: \mathbb{C}-\{0\} \mapsto \mathbb{R}-\{0\}$, then $x \in \mathbb{C} - \{0\}$, and $\phi(x) \in \mathbb{R} - \{0\}$, then $\vert x \vert = \vert \phi(x) \vert$. Now note that the element $i \in \mathbb{C} - \{0\}$ has order $4$. However, no element in $\mathbb{R}-\{0\}$ has order $4$. Hence, no isomorphism can exist. Hence, the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.    Thanks
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I saw  and it states that the Lie algebra of the special linear group $SL_n(\mathbb C)$ is the group of traceless matrices $\mathfrak{sl}_n$.  I know the definition of a Lie algebra: given a matrix group $G \subseteq GL_n(\mathbb K)$ its Lie algebra is defined to be the tangent space at $I$.  From this definition it is immediately clear what dimension the Lie algebra has to have.  What is unclear to me is:      Assuming I don't already know the answer, how do I arrive at the insight that this tangent space is traceless matrices?   I skipped through Tapp's book in search of hints and found that the exponential map is a homomorphism from the Lie algebra into the Lie group. I also found a lemma that states that   $$ \det e^A = e^{\mathrm{trace}(A)}$$  That is unfortunately only half-useful: If I already know what the Lie algebra is then this lets me prove that the image of the Lie algebra under the homomorpism defined by the exponential map is indeed in $SL_n$.   So it's kind of not so useful. Surely there must be geometric insight?
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My professor keeps mentioning the word "isomorphic" in class, but has yet to define it... I've asked him and his response is that something that is isomorphic to something else means that they have the same vector structure. I'm not sure what that means, so I was hoping anyone could explain its meaning to me, using knowledge from elementary linear algebra only. He started discussing it in the current section of our textbook: General Vector Spaces.  I've also heard that this is an abstract algebra term, so I'm not sure if isomorphic means the same thing in both subjects, but I know absolutely no abstract algebra, so in your definition if you keep either keep abstract algebra out completely, or use very basic abstract algebra knowledge, that would be appreciated.
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Show that $D_{12}$ is isomorphic to $D_6 \times C_2$, where $D_{2n}$ is the dihedral group of order $2n$ and $C_2$ is the cyclic group of order $2$.   I'm somewhat in over my head with my first year groups course. This is a question from an example sheet which I think if someone answered for me could illuminate a few things about isomorphisms to me.   In this situation, does one use some lemma (that the direct product of two subgroups being isomorphic to their supergroup(?) if certain conditions are satisfied)? Does $D_{12}$ have to be abelian for this? Do we just go right ahead and search for a fitting bijection? Can we show the isomorphism is there without doing any of the above?  If someone could please answer the problem in the title and talk their way through, they would be being very helpful.  Thank You.
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I'm a beginner of QFT. Ref. 1 states that      [...] The Lorentz group $SO(1,3)$ is then essentially $SU(2)\times SU(2)$.    But how is it possible, because $SU(2)\times SU(2)$ is a compact Lie group while $SO(1,3)$ is non-compact?   And after some operation, he says that the Lorentz transformation on spinor is complex $2\times2$ matrices with unit determinant, so Lorentz group becomes  $SL(2,\mathbb{C})$. I'm confused about these, and I think there must be something missing.   References:   L.H. Ryder, QFT, chapter 2, p. 38.
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Let $G$ be an abelian group such that the Schur multiplier $M(G)=\mathbb{Z}_p\times \mathbb{Z}_p$.     Can we say that $G$ has a subgroup isomorphic to $\mathbb{Z}_p\times \mathbb{Z}_p.$   I have seen in many groups this relation holds, what can we say in general? I don't know that how to prove this or contradict. Please help me.
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In general, it is not too hard to find isomorphisms between two groups when their order is relatively low. However, as their orders grow, it becomes increasingly irritating to write down their entire Cayley tables and such. Is there a set of tricks that is generally useful when trying to prove that two groups are actually isomorphic? After all, it usually seems easier to prove that they aren't, as you just need to point out one property that doesn't correspond...  Example: in Armstrong's Groups and Symmetry, it is asked to show that the dihedral group of order 8 and the subgroup of S4 generated by (1234) and (24) are isomorphic. It is easy to send D4's "single-rotation" element r to S4's (1234) and D4's "flipping" element s to S4's (24), as they are all part of the generating set and their orders coincide, but what is the way to go from here? Also, how far should one go in showing the isomorphism - might pointing out the correspondence in generating elements even be enough?
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Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $D_6$ (the group with $x^3=1$, $y^2=1$ and $xy=yx^2$).  I'm not really sure how to express the elements of $\operatorname{Aut}(C_2 \times C_2)$. Would it be sufficient to show the elements of $\operatorname{Aut}(C_2 \times C_2)$, find their order and show they bijectively map to every element of $D_6$ and satisfy $xy=yx^2$?
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Let $V$ be a finite dimensional vector space (of dimension n) over a field $F$. I need to show that $V$ is isomorphic to $F^n$ as abelian groups. However, I don't really understand what does "isomorphic as abelian groups" mean, my (poor) attempt to solve this was:  Let $f:V\to F^n$, $f((v_1,v_2,\ldots,v_n)):=v_1+\ldots+v_n$, clearly $f$ is a group homomorphism, but it is not biyective since, for example, in the case where $dim(V)=2$ and $F=\mathbb{R}$ we have $f((2,-2))=0=f((3,-3))$, I've also tried it by defining $f$ to be the product of a vector's entries but obviously it didn't work so I'm stuck!   I think this should be an easy problem but since I don't quite understand it, it's giving me problems. Thank you for your help.
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Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\exp :\mathfrak{g}\rightarrow G$ be the exponential map.  In his blog,  that if a Lie group is not simply-connected, then $\exp$ will not be injective.  Conversely, is it true that if a Lie group is simply-connected, then $\exp$ is injective?  If not, what is a counter-example?
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Show that up to isomorphism there exist only two groups of order 10: $C_{10}$ and $D_{10}$. I need some help on this question. I only know the basic definitions of isomorphism. Any hints?
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I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics?  The application of Lie groups and their representations is an enormous field, with vast implications for physics with respect to such things as unification, but I what specifically made these groups of physical importance and why there study is useful.  I have just started studying these two groups in particular, but from a mathematical perspective, I'd very much appreciate understanding some sort of physical motivation.
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I would like to do  (i) but before I can even attempt it I need to be clear about his terminology.  Exercise 6  Show that the Lie algebra $gl_n(\mathbb{C})$ of the general linear group $GL_n(\mathbb{C})$ can be identified with the space $M_n(\mathbb{C})$ of $n \times n$ complex matrices, with the Lie bracket $[A,B] := AB - BA$.  When he writes "identified", does he just mean there is an isomorphism? So the exercise would be to find an explicit isomorphism from $M_n(\mathbb{C})$ to $gl_n(\mathbb{C})$? Is that correct?  Many thanks for your help.
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I'm looking for an example of of two isomorphic abelian groups, which are not isomorphic $R$-modules for some ring $R$.   I suppose we can just the same abelian group $M$ twice, and use a different operation $R\times M \rightarrow M$ so the $R$ modules aren't isomorphic.  I can't think of such a group $M$ and ring $R$ to make this possible, though. Any ideas? Thanks.
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It is well know that $SO(3)$ is a 3-dimensional Lie group. My question is  $SO(3)$ don't has 2-dimensional Lie subgroups?
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I am reading the notes of Victor Kac (Introduction to Lie Algebras). After Cartan's theorem there is an  application, the classification  (up to isomorphism) of 3-dimentional Lie algebras.  We use the Cartan's theorem to write   $$ \mathfrak{g} = \bigoplus_{\lambda\in \mathbb{F}}\mathfrak{g}^{a}_{\lambda} $$  where $a \in \mathfrak{g}$ is a regular element and $\mathfrak{h}=\mathfrak{g}_{0}^{a}$ is a subalgebra de Cartan.  I have proven when $\mbox{rank }\mathfrak{g}=\dim\mathfrak{h}=1$ one has  3 possibilities for $\mathfrak{g}=\mbox{span}\{a,b,c\}$:  \begin{eqnarray} 1 &amp;-&amp; [a,b]=b, [a,c] = c + b, [b,c] = 0\\ 2 &amp;-&amp; [a,b]=b, [a,c] = \lambda c, [b,c] = 0, \mbox{ for a non null $\lambda$} \\ 3 &amp;-&amp; [a,b]=b, [a,c] = -c, [b,c] = a \end{eqnarray}  So my question is: How proof that these 3 Lie algebras are non isomorphic?  I would be happy with any help!