Question

5) a) Let H = h(1 2)i and K = h(1 2 3)i be subroups of S3. Check that S3 = HK. Is S3 the internal
direct product of H and K? Justify your answer.
b) Let s = 1 2 3 4 5 6 7
2 4 5 6 7 3 1and t = 1 2 3 4 5 6 7
3 2 4 1 6 5 7be elements of S7.
i) Write both s and t as product of disjoint cycles and as a product of transpositions,
ii) Find the signatures of s and t.
iii) Compute ts

Answer 1

Answer on Question #44356 – Math - Abstract Algebra

Problem.

5) a) Let $H = h(12)i$ and $K = h(123)i$ be subroups of S3. Check that $S3 = HK$. Is S3 the internal direct product of H and K? Justify your answer.

b) Let $s = 1234567$

2456731 and $t = 1234567$

3241657 be elements of S7.

i) Write both $s$ and $t$ as product of disjoint cycles and as a product of transpositions,

ii) Find the signatures of $s$ and $t$.

iii) Compute $ts$

Remark.

The statement isn't correctly formatted. I suppose that the correct statement is

"5) a) Let $H = \langle (12) \rangle$ and $K = \langle (123) \rangle$ be subroups of $S_3$. Check that $S_3 = HK$. Is $S_3$ the internal direct product of $H$ and $K$? Justify your answer.

b) Let $s = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 4 & 5 & 6 & 7 & 3 & 1 \end{pmatrix}$ and $t = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 4 & 1 & 6 & 5 & 7 \end{pmatrix}$ be elements of $S_7$.

i) Write both $s$ and $t$ as product of disjoint cycles and as a product of transpositions,

ii) Find the signatures of $s$ and $t$.

iii) Compute $ts$

Solution.

a) The elements of $H$ are $\{e, (12)\}$.

The elements of $K$ are $\{e,(123),(132)\}$

There 6 elements in $S_3$ each of it could be presented as product elements from $H$ and $K$.

$e = ee$;

$(12) = (12)e$;

$(13) = (12)(132)$;

$(23) = (12)(123)$;

$(123) = e(123)$;

$(132) = e(132)$.

b)

i) $s = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 4 & 5 & 6 & 7 & 3 & 1 \end{pmatrix} = (1246357) = (12)(24)(46)(63)(35)(57)$.

$t = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 4 & 1 & 6 & 5 & 7 \end{pmatrix} = (134)(56) = (13)(34)(56)$.

ii) $\operatorname{sgn}(s) = (-1)^6 = 1$ and $\operatorname{sgn}(t) = (-1)^3 = 1$.

iii) $ts = (134)(56)(1246357) = (12)(36457)$.

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