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Bookmark or Share- Subgroups of $\\mathbb{Z}$ - Mathematics Stack ExchangeI am trying to understand subgroups.I know a given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
- Find the subgroups of A4 - Mathematics Stack Exchange$\begingroup$ A subgroup of order 3 is isomorphic to $\Bbb Z_3$, so it is generated by an element of order 3. To identify the subgroups of order 3 it suffices to identify the elements of order 3. $\endgroup$ –
- abstract algebra - Is an ideal also a normal subgroup? - Mathematics ...Since a ring is moreover an abelian group under addition, every subgroup is normal. EDIT: To explicitly answer your questions: 1) Since a subring is also a subgroup and addition is abelian, something must first be a normal subgroup in order to be an ideal. 2) Yes, well-definedness under addition does technically require being a normal subgroup.
- Finding subgroups of $S_5$ with specific ordersFor 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 $(12345)$.
- 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. $\endgroup$ –
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