### Questions (102)

 5 If a group $G\leq S_{13}$ has an element of order $40$, then $G$ has a normal non-trivial subgroup. 4 How many different homomorphisms $\varphi : \mathbb{Z}_{18} \to \mathbb{Z}_{24}$ so $|Im\varphi| =6$ 4 Group of order $5^k\cdot 8$ has normal subgroups of order $5^{k},5^{k}\cdot2,5^{k}\cdot4$ 3 Find $\tau$ so $\tau^5 = \sigma$ when $\sigma = (1,4,7)(2,5,3,9)\in S_9$ 3 Understanding the difference between two theorems related to $|G\,:\,K|=|H\,:\,K|\cdot|G\,:\,H|$.

### Reputation (1,009)

 +10 Proving that function is continuous and differentiable at $x=0$ +10 Proving that set $U$ is subspace of space $V$ +10 Checking the series $\frac{a}{b}-\frac{a(a+1)}{b(b+1)}+\frac{a(a+1)(a+2)}{b(b+1)(b+2)}-\cdots$ +10 Checking the series $\sum_{n=1}^{\infty}\frac{e^n+n^4}{(-3)^{n^2-n+1}+\ln(n+1)$

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### Tags (35)

 0 group-theory × 45 0 combinatorics × 7 0 abstract-algebra × 32 0 graph-theory × 4 0 calculus × 18 0 elementary-number-theory × 4 0 permutations × 10 0 matrices × 3 0 linear-algebra × 9 0 modular-arithmetic × 3