# Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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### Number of subgroups of groups with prime power order

Let $G$ be a group of order $p^2$ and put $\mathcal A=\{U\leq G, \#U=p\}$. What is $\#\mathcal A$? If $G$ is cyclic, then $G$ is generated by some element $x$ of order $p^2$. It seems like there ...
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### When is showing any elements in a subset of a group is a sufficient condition for a subgroup. [on hold]

Let G/N be an abelian factor group. To show that H is a subgroup of N, it suffices to show that H is a subset of N. Why is it not necessary in this case to demonstrate that the group axiom holds or ...
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### Why is identity element required for groups?

I would like to know the necessity for having an identity element for every group. I know the meaning of an identity element.
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### Why is SU(2) not the same group as T3?

Why is $S^3 = SU(2)$ not the same group as $S^1 \times S^1 \times S^1$? It seems that, the $T^3$ torus is abelian, wheras $S^3$ is not. Is that enough? Problem 2.6.5 from John Stillwell, Naive ...
### Showing $NH$ is a subgroup of $G$
Question : If $N$ is a normal subgroup of $G$ and $H$ is a subgroup of $G$, prove that $NH$ is a subgroup of $G$. Thread is constructed on a mobile so I will attempt to be as succinct as possible. ...
### A homomorphism $f:S_{n} \rightarrow \mathbb{R}$, prove that $f$ must be trivial.
A homomorphism $f:S_{n} \rightarrow \mathbb{R}$, prove that $f$ must be trivial, i.e., $\ f(\sigma) = 0 \ \forall \sigma \in S_{n}$. I started off by thinking that it was something to do with the ...