What do I need to show that a subset of a group is a subgroup? 1) The identity from the group is the identity for the subgroup and is in the subgroup 2) The group is closed under inversion (group operation with inverse element) and that the inverse for the group is the inverse for the subgroup 3) The group is closed under the group product If I am wrong, please correct me with the proper approach
abstract algebra - 3 different subgroup tests. When to use each? are . . . $\begingroup$ I also noticed that all the 3 subgroup tests proofs involved using the one-step subgroup test I guess I will try multiple of problems and try all 3 and hopefully I would see which will fit the nature of the groups and subgroups $\endgroup$
Subgroup generated by a set - Mathematics Stack Exchange A subgroup generated by a set is defined as (from Wikipedia):More generally, if S is a subset of a group G, then , the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S; equivalently, is the subgroup of all elements of G that can be expressed as the finite product of elements in S and
group theory - What exactly a proper subgroup means? - Mathematics . . . The question seems very simple, but it's confusing me as the term 'proper subgroup' has different definations in different reference books I read in galian(7th edition) that the subgroup of G Other than G itself are proper But some books, and some of my teachers believes that {e} is also an improper subgroup of G
How can we find and categorize the subgroups of Also when H is a subgroup of R looking at the structure of the cosets of R H eg for H any of the Z subgroups we get R H homomorphic to [0,1) or the circle For H one of the Q subgroups it is more complex and I currently don't have ideas on the larger subgroup cosets I am not clear how "big" a subgroup H can get before it becomes the whole
Finding subgroups of $S_5$ with specific orders For a subgroup of order $20$ we can take an element of order $5$, which exists by Cauchy, i e , a $5$-cycle $(12345)$ and a $4$-cycle $(2354)$ to obtain a subgroup of order $20$ Here the $4$-cycle normalizes the subgroup generated by $(12345)$
Subgroups of dihedral group - Mathematics Stack Exchange Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers
How to find all subgroups of - Mathematics Stack Exchange $\begingroup$ The subgroup generated by 1 and 1 2 is generated by 1 2 alone There are two cases: there is a smallest positive element in this subgroup (which you must show generates the group), or there is none In the second case, look at the set of denomiators inside the integers and try to find its structure $\endgroup$ –
Subgroups of a direct product - Mathematics Stack Exchange Until recently, I believed that a subgroup of a direct product was the direct product of subgroups Obviously, there exists a trivial counterexample to this statement I have a question regarding