Question

How to calculate the automorphisms group of nonzero rational number multiplicative group?

score 219 · Answer 1 · 2026-04-17 15:31:10

How to calculate the automorphisms group of nonzero rational number multiplicative group?

Published on 17 Apr 2026 - 15:31

score 102 · Answer 2 · 2026-04-14 04:55:21

Let $f:\Bbb{R^\times}\to\Bbb{R^\times}$ be an isomorphism. Show that $f$ takes $\Bbb R^{>0}$ to $\Bbb R^{>0}$, and $\Bbb R^{<0}$ to $\Bbb R^{<0}$.

score 2.5k · Answer 3 · 2026-05-10 16:05:44

2.5k

Views

Is $\Bbb R/\Bbb Z$ isomorphic to $\Bbb R/2\Bbb Z$?

Published on 10 May 2026 - 16:05

score 440 · Answer 4 · 2026-04-12 02:05:01

440

Views

Isomorphism between direct products

Published on 12 Apr 2026 - 2:05

score 102 · Answer 5 · 2026-04-12 09:16:13

102

Views

Is $ \{ x+xi \ | \ x \in R \}$ isomorphic to a straight line through the origin in $R^2$?

Published on 12 Apr 2026 - 9:16

score 91 · Answer 6 · 2026-04-11 20:30:06

91

Views

Are the groups $\mathbb{C}$ and $\mathbb{R}^2$ isomorphic under addition?

Published on 11 Apr 2026 - 20:30

score 589 · Answer 7 · 2026-05-10 17:16:11

589

Views

If group homomorphisms preserve inverses, why aren't they just isomorphisms?

Published on 10 May 2026 - 17:16

score 87 · Answer 8 · 2026-04-15 10:22:35

87

Views

Prove that $\mathbb{R}^\times\times\mathbb{R}^\times\ncong\mathbb{C}^\times$

Published on 15 Apr 2026 - 10:22

score 946 · Answer 9 · 2026-04-17 03:09:47

946

Views

Verify that the following is an isomorphism $T$:$P_1$ → $R^2; T[p(x)]=[p(0),p(1)]$

Published on 17 Apr 2026 - 3:09

score 2.1k · Answer 10 · 2026-04-10 05:48:13

2.1k

Views

In general, what techniques can be used to show that 2 groups are not isomorphic?

Published on 10 Apr 2026 - 5:48

score 37 · Answer 11 · 2026-04-16 13:42:03

37

Views

Show that $f_{|\,f[G]}$ is an isomorphism where $f(g)=g^2$ (no elements of order 4)

Published on 16 Apr 2026 - 13:42

score 1.6k · Answer 12 · 2026-05-10 15:39:19

1.6k

Views

Elliptic curve group of $y^2 = x^3 + 2x + 3$ over $\mathbb{F}_5$

Published on 10 May 2026 - 15:39

score 44 · Answer 13 · 2026-05-10 17:13:32

44

Views

$\mathbb{Z}^k/A\mathbb{Z}^k \cong\mathbb{Z}^k/A'\mathbb{Z}^k$

Published on 10 May 2026 - 17:13

score 329 · Answer 14 · 2026-04-12 20:31:56

329

Views

Show that $\mathbb{Z}$ is isomorphic to $n\mathbb{Z}$.

Published on 12 Apr 2026 - 20:31

score 597 · Answer 15 · 2026-04-15 06:10:47

597

Views

Show that $S$ is the internal direct product isomorphic to $\Bbb{Z}\times \Bbb{Z}$.

Published on 15 Apr 2026 - 6:10

List Question

How to calculate the automorphisms group of nonzero rational number multiplicative group?

Let $f:\Bbb{R^\times}\to\Bbb{R^\times}$ be an isomorphism. Show that $f$ takes $\Bbb R^{>0}$ to $\Bbb R^{>0}$, and $\Bbb R^{<0}$ to $\Bbb R^{<0}$.

Is $\Bbb R/\Bbb Z$ isomorphic to $\Bbb R/2\Bbb Z$?

Isomorphism between direct products

Is $ \{ x+xi \ | \ x \in R \}$ isomorphic to a straight line through the origin in $R^2$?

Are the groups $\mathbb{C}$ and $\mathbb{R}^2$ isomorphic under addition?

If group homomorphisms preserve inverses, why aren't they just isomorphisms?

Prove that $\mathbb{R}^\times\times\mathbb{R}^\times\ncong\mathbb{C}^\times$

Verify that the following is an isomorphism $T$:$P_1$ → $R^2; T[p(x)]=[p(0),p(1)]$

In general, what techniques can be used to show that 2 groups are not isomorphic?

Show that $f_{|\,f[G]}$ is an isomorphism where $f(g)=g^2$ (no elements of order 4)

Elliptic curve group of $y^2 = x^3 + 2x + 3$ over $\mathbb{F}_5$

$\mathbb{Z}^k/A\mathbb{Z}^k \cong\mathbb{Z}^k/A'\mathbb{Z}^k$

Show that $\mathbb{Z}$ is isomorphic to $n\mathbb{Z}$.

Show that $S$ is the internal direct product isomorphic to $\Bbb{Z}\times \Bbb{Z}$.

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