Solving Equations in Groups Let a, b, c, and x be elements of a group G. In each

Question

Solving Equations in Groups Let a, b, c, and x be elements of a group G. In each of the following, solve for x in terms of a, b, and c. Example Solve simultaneously: x^2 = b and x^5 = e From the first equation, b = x^2 Squaring, b^2 = x^4 Multiplying on the left by x, xb^2 = xx^4 = x^5 = e. Multiplying by (b^2)^-1, xb^2(b^2)^-1 = e(b^2)^-1. Therefore, x = (b^2)^-1. Solve: 1 axb = c 2 x^2 b = xa^-1 c Solve simultaneously: 3 x^2a = bxc^-1 and acx = xac 4 ax^2 = b and x^3 = e 5 x^2 = a^2 and x^5 = e 6 (xax)^3 = bx and x^2 a = (xa)^-1

Explanation / Answer

1) axb = c

multiplying both sides by a^-1 b^-1

axb (a^-1 b^-1 ) = c a^-1 b^-1

hence, x = c a^-1 b^-1

2) x^2 b = x a^-1 c

multiplying both sides by x^-1

x^-1 (x^2 b ) = x^-1( x a^-1 c )

x b = a^-1 c

x = a^-1 b^-1 c

3) x^2 a = bxc^-1 and acx = xac

multiplying both sides by x^-1

x^-1 (x^2 a ) = x^-1 (bxc^-1 )

xa = bc^-1

multiplying both sides by c

xa(c) = bc^-1 (c)

xac = b

acx = b

x = a^-1 b c^-1

4) ax^2 = b and x^3 = e

multiplying both sides by x

ax^2 (x) = bx

ax^3 = bx

x^3 = e

ae = bx

x = ab^-1

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