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subgroup    音标拼音: [s'ʌbgr,up]
n. 小群,隶属的小组织,子群

小群,隶属的小组织,子群

subgroup
子群

subgroup
n 1: a distinct and often subordinate group within a group
2: (mathematics) a subset (that is not empty) of a mathematical
group

Subgroup \Sub"group`\, n. (Biol.)
A subdivision of a group, as of animals. --Darwin.
[1913 Webster]


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  • 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
  • Understanding how to prove when a subset is a subgroup
    Lemma 3 4 Let $(G ,*)$ be a group A nonempty subset $H$ of $G$ is a subgroup of $(G,*)$, iff, for every $a, b\in H$, $a*b^{-1}\in H$
  • What is the difference between a Subgroup and a subset?
    A subgroup is a subset which is also a group of its own, in a way compatible with the original group structure But not every subset is a subgroup To be a subgroup you need to contain the neutral element, and be closed under the binary operation, and the existence of an inverse
  • 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
  • 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 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
  • Difference between conjugacy classes and subgroups?
    As others said subgroup has all the properties of Group But conjugacy classes are just the set, but created with conjugacy and are equivalence relation Intuitively conjugacy is, looking the same thing with different perspective For ex take ${D_6}$, a hexagon and say r=clockwise rotation and f=horizontal reflection
  • Whats an easy way of proving a subgroup is normal?
    If your subgroup has index 2, then it is always normal (because whether you consider left or right cosets, there are only these 2: the subgroup itself, and the rest of the elements) Another way (maybe the best way) is to show that the subgroup is the kernel of a homomorphism having the group as its domain
  • 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





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